This article was downloaded by: 10.3.98.104 On: 27 Sep 2021 Access details: subscription number Publisher: CRC Press Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK

The Weibull Distribution A Handbook Horst Rinne

Related distributions

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781420087444.ch3 Horst Rinne Published online on: 20 Nov 2008

How to cite :- Horst Rinne. 20 Nov 2008, Related distributions from: The Weibull Distribution, A Handbook CRC Press Accessed on: 27 Sep 2021 https://www.routledgehandbooks.com/doi/10.1201/9781420087444.ch3

PLEASE SCROLL DOWN FOR DOCUMENT

Full terms and conditions of use: https://www.routledgehandbooks.com/legal-notices/terms

This Document PDF may be used for research, teaching and private study purposes. Any substantial or systematic reproductions, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC o ausof values for qainfrtepoaiiydniyfunction density probability the for equation nwa anr efis ics Scin31 o h W the how 3.1) (Section discuss first We manner. what in mohdi natmtt osrc kwdcniuu densi continuous skewed construct probabilities to the hypergeom attempt and an binomial skewed in the smoothed point starting his as took W The the incorporate distributions. either of that families distributions familiar or systems well–known the of ra ait fdsrbto oesta aebe derived Sections been have (in that Then models 3.2). distribution (Section of way variety some great in it to related are . ytm fdsrbtosadteW the and distributions of Systems 3.1 th in presentation the 2 , level. Chapter probabilistic in As or other. the or way one in itiuin a ecasfidit aiiso ytm suc systems or families into classified be may Distributions W the distributions other which explores chapter This distributions Related 3 iso apigdsrbtosdsoee usqetyby subsequently discovered theoreti distributions a sampling provided of it lies Secondly, applications. of many number in small data a involving — Fi representations reasons. mathematical two for development K significant by a developed was troduction was distributions of system oldest The .. P 3.1.1 possible. approximations as distributions provide empirical to or designed served been have families Such 1 • • • ugse edn o hsscin B section: this for reading Suggested J OHNSON hr h aestructure. same and/or the design share common a to according constructed been have and/or properties special same the have EARSON /K r OTZ nieterne iiigagmn ugssacomparable a suggests argument limiting A range. the inside P r /B o h yegoercdsrbto aif h differenc the satisfy distribution hypergeometric the for system ALAKRISHNAN f ′ ( x P = ) r − 19,Catr4,O 4), Chapter (1994, P d f r d − ( ARNDORFF x x 1 ) = = b 0 b 0 + ( ( –N r + x b − 1 − IELSEN b EIBULL r 1 RD a a x + ) EIBULL ) 17) P (1972), ARL P + f b r EIBULL EIBULL 2 ( b 17) E (1978), P x a rmwr o aiu fami- various for framework cal r 2 rmteW the from itiuina pca aeor case special a as distribution httemmeso family a of members the that h EARSON sl,tesse ile simple yielded system the rstly, ) 2 P x scatri ntetheoretical the on is chapter is efidnswl edt some to lead will findings se EARSON 2 .. o331)w rsn a present we 3.3.10) to 3.3.1 aaees—frhistogram for — parameters ti itiuin,wihhe which distributions, etric itiuini eae oand to related is distribution EARSON yfntos entdthat noted He functions. ty oa ieavreyo ob- of variety a wide as to . itiuinfisit some into fits distribution LDERTON distribution n tes P others. and /H EIBULL rud19.Isin- Its 1895. around ARTLEY /J equation e OHNSON distribution differential (1972). EARSON (1969), 1 (3.1) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC uhadarmepaiigta raweew n h W the find we where area that emphasizing diagram a such ai diagram ratio rvd opeetxnm ftesse htcnb depicte be can that system the of taxonomy complete a provide e etr fteP the of feature key A cases. special as follow types “transition” more Ten eepesdepiil ntrso h orparameters four the of terms in explicitly expressed be h w oet ratios, moments two the h aiu ye rfmle fcre ihnteP the within curves of families or types various The om fsltost 31.Teeaetremi distributi P main by three VI are and There IV I, (3.1). types to solutions of forms h solutions The W the and distributions of Systems 3.1 eoiao f(3.1): of denominator • • • • • • h iefrtp ,sprtn h ein ftp–Iadtyp and type–VI of regions the by separating given V, type for line The tsprtstergoso h yeIadtp–Idistribut type–VI and type–I the of regions the separates It h iefrtp I ( III type for line The rsae otherwise: stated or h ii o l itiuin sgvnby given is distributions all for limit The yeV ihDF with VI type complex; are roots two the when results eut hntetorosaera ihtesm in (The sign. same kind the second with the real of are roots two the when results yeI ihDF with IV type (The signs opposite with kind real first are roots two the when results yeIwt DF with I type f with ( x so yeI.); type of is ) f r h est ucin fthe of functions density the are ( β x EARSON 1 1+ (1 = ) β nteasis and abscissa the on EARSON 1 f so yeVI.) type of is β ( ( am distribution gamma β 2 x f 2 1+ (1 = ) ≤ ( 3) + x ytmi httefis ormmns(hnte xs)may exist) they (when moments four first the that is system β β x = ) + 1 hc r eeae yteroso h udai nthe in quadratic the of roots the by generated are which , 2 1 2 ) 2 − EIBULL x β = = m (4 4 = m 1 . x exp 2 β ) 2 1+ (1 µ µ µ µ m β 2 2 2 3 4 3 2 1 + 3 = 2 1  distribution β β − (1 2 2 − (kurtosis) (skewness) x − β nteordinate. the on − ) ν 1 sgvnby given is ) − 3 x tan 1 = m . P β ) EARSON 5 1 m 1 EARSON , β (2 ) 1 2 , 1 ( ( , 0 , x . ,b a, , ) ≤ − β

EIBULL 2 1 , n ntesse,designated system, the in ons < x 0 − ytmcrepn odistinct to correspond system b , ≤ −∞ system 3 naso–called a in d x 1 ions. i.3/1 Fig. ∞ β F eadsrbto fthe of distribution beta and ≤ 1 < x < , distribution. − or – 1 –Vdsrbtos is distributions, e–IV . , b 6) 2 f(.) nturn, In (3.1). of ) eadistribution beta hw ealof detail a shows ∞ , moment– 99 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC iue3/1: Figure 100 When the region. to type–VI corresponding line a by and edn ntesaeparameter shape the on pending ehv lomre ydt h oiin ffu te specia other four of positions the dots by marked also have We • • • • rnho h W the of branch h oe rnho h W the of branch lower the yeIeteevledistribution value extreme type–I xoeta distribution exponential omldistribution normal uniform r—sle for solved — or β 2 spotdover plotted is bution oetrtodarmfrteP the for diagram ratio Moment or etnua distribution rectangular β 2 = EIBULL β 2 β 3 by — 1 ( o W a for − β 1 –distribution–line, 16 0; = EIBULL ( β c − onra distribution lognormal 1 hsfnto a etxat vertex a has function This . EIBULL 13 4; = β β 2 β –distribution–line, 1 3) = 1 − ( − β β EARSON 2 1 ( 32 itiuinw e aaercfnto de- function parametric a get we distribution 2 β , 9) = ≈ p 1 0; = 1 4+ (4 . 2986; ttecosn ftegmaln and gamma–line the of crossing the at ytmsoigteW the showing system β β 2 1 ) 1 = β 3 2 hc opeeyflsit the into falls completely which
5 = . , 8) distributions: l 0 , . 4) ≤ tteedo h upper the of end the at β ( eae distributions Related 3 1 β 1 < β , 32 2 EIBULL ) . ≈ (0 , distri- 2 . 72) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC eeaie yitouigalcto aaee n scale a and parameter location a introducing by generalized ihDF with ti eurdthat required is It n oet bu h rgngvnby given origin the about moments and iet sflrneo au fskewness, of value of range useful a to rise inlfrs aho hc ie iet aiyo Dswith CDFs of family a to rise gives which of each forms, tional function ihrsett h W the to respect With families. such twelve listed naaou otedfeeta qain(.)ta generat that (3.1) equation differential the to analogue an hie of choices neet t D is CDF Its interest. uvs CDF A curves. The B 3.1.2 distribution. lognormal the n nta iea on hc ak h yeIeteevalu extreme type–I the when tribution marks which point a at line that on ing orsodn to corresponding W the and distributions of Systems 3.1 yeV eina on with cr point It a line. at type–V region the type–VI to parallel approximately extends and ytm For system. W the that sees ( uinwith bution est ucin,t rqec aa hsaodn h pro percentil the or avoiding probabilities when thus encountered are data, which frequency tion to functions, density xed prxmtl aallt h yeII(am)lin (gamma) type–III the at to parallel approximately extends W β 1 EIBULL ( 2 β , β B 1 ugse edn o hsscin R section: this for reading Suggested 2 β , URR ) 2 ≈ g (4 = ) URR ( g itiuinwith distribution ,y x, (1 system E ( c ,y x, < c . 2928 1 = X β ) y EIBULL system . 1 utb oiiefor positive be must ) 0 r c := > k c
≥ , 3 eeaevrossolutions various generate .TeW The ). 9 , . ≈ = (see 6023 . 5 0 F 0) . . 3 µ 4) 5 ( EIBULL . 2 x n for and , r ′ orsodn oteepnnildsrbto =W (= distribution exponential the to corresponding itiuinde o eogt nyoefml fteP the of family one only to belong not does distribution 6023 4 B ) n nifiielwrbac vldfor (valid branch lower infinite an and = h W the F o h orhmmn,adthus and moment, fourth the for d UR d nteB the in x y > c ( k EIBULL x f ihafiieuprbac vldfor (valid branch upper finite a with Γ = 1 = ) ( β  itiuinteB the distribution 14) t uuaiedsrbto ucin,rte tha rather functions, distribution cumulative fits (1942)) x EIBULL 1 β y 3 k = ) . 1 (1 ≈ 6023 URR − ODRIGUEZ prxmtl rae hn10i ilcoeyresemble closely will it 1.0 than greater approximately EIBULL − − 0 0 iefor line x c k r c .  1+ (1 5 ≤ y ytmhst aif h ifrnilequation: differential the satisfy to has system ilcoeyrsml h P the resemble closely will itiuinle anyi h yeIrgo and region type–I the in mainly lies distribution ) n hnmvstwr h onra ie end- line, lognormal the toward moves then and Γ c y g  α − distribution 1 ( x 17) T (1977), 1 3 ≤ r ,y x, c F > c c 1+ (1 = 1 + ) ( 1 k x ) ± ) and y , URR ; .  hs a ecasfidb hi func- their by classified be can These . 3 √ x ,c > k c, x, . ADIKAMALLA c 6023 β ) x := 1 − yeXIdsrbto so special of is distribution type–XII Γ( n kurtosis, and , ntespotof support the in k F − k saeeautdfo P from evaluated are es steP the es rgntsi h yeIregion type–I the in originates 1 ( se h yeIIln nothe into line type–III the osses β parameter. 1) + ni h w ie intersect lines two the until e x 2 lm fnmrclintegra- numerical of blems 0 ) oeit hsfml gives family This exist. to , nteB the in , , itiuin ec,the Hence, distribution. e < c (1980a). for > c EARSON EARSON 3 r. > k c URR α . 6023) 3 4 . F = 6023) ytm B system. EIBULL ( β x ytm The system. yeV dis- type–VI n easily One . 2 ) tmybe may It . Different . nigat ending EARSON EARSON (3.3b) distri- (3.3c) (3.3a) (3.2) URR 101 n Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h pe on nFg / srfre oa lwrbud and bound” “lower for as equations to parametric referred The is text. furtherm 3/2 Fig. and in skewness, bound negative upper the as well as positive showing iue3/2: Figure where hra h hr n orhmmn obntoso h P the of combinations moment fourth and third the Whereas 102 lap aiyw s nte yeo oetrtodarmwith diagram moment–ratio of type another use we family 3 3 ehv noelpigwe okn tteB the at looking when overlapping an have we h aei refrteJ the for true is same The bution oetrtodarmfrteB the for diagram ratio Moment β 2 = λ j Γ p =Γ := 3 OHNSON ( β k 1 )  λ = 4 j c − Γ aiis(see families 1 + √ 2 Γ 4 ( β k  1 )  2 λ Γ( and  ( Γ Γ( k 3  k ) − k ) k λ β URR et 3.1.3 Sect. ) λ 3 Γ( 3 2 − λ 2 λ are 2 − 1 j c yeXIfml n h W the and family type-XII − k URR  Γ( 6 + λ ) λ 1 2 λ ;  ). 1 2 2 3  aiis o eitn h type–XII the depicting For families. j / λ 3 2 / k 1 1 = 2 ) 2 + λ α EARSON , 2 3 r ti piedw.Thus down. upside is it ore 2 λ λ , ovreyi h following the in conversely = 1 2 1 3 3 , − ± 4 . 3 √ eae distributions Related 3 aiisd o over- not do families λ β 1 4 1 sasis,thus abscissa, as EIBULL (3.3d) distri- (3.3e) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ( ec,teW the Hence, aiycnas eepanda olw,satn rm(3.3a from starting follows, as explained be also can family efial eto the mention √ finally We 34,)t 29)ad(.0) h dnicto ftelow the of identification The (2.101). and (2.93) to (3.4a,b) obnd opoieoedsrbto orsodn oeac to corresponding distribution one provide si do such combined, No J distributions. but of available, family corresponding the of ber hslwrbudi dnia oteW the to identical is bound lower This where tedpit ihnegative with end–points at ein R region. which ntepositive the In h rnfraino ait onraiyi h ai ft of basis the is normality to variate a of transformation The J 3.1.3 g ihteP the with ogy efudsc ht o n osbepi fvalues of pair possible any for that, such found be rwo nacetRmngle.As galley. Roman ancient an of prow . ytm fdsrbtosadteW the and distributions of Systems 3.1 nacutrcokiedrcintwr h i fte’prow the of tip the toward direction counter–clockwise a in ie emnt tedpit ihpositive with end–points at terminate lines R − ODRIGUEZ β 1 1 . 14 afpaetebudi ie for given is bound the half–plane k , 5 1 = ODRIGUEZ . OHNSON 4) Pr " and to tts Tetp–I B type–XII “The states: EIBULL X √ EARSON OHNSON β √ ≤ > c 1 β  afpanti on orsod oB to corresponds bound this half–plain system 1 17)gvstefloigprmti qain fteend– the of equations parametric following the gives (1977) 4 β , k 1 aiyi h iiigform limiting the is family k  a enpoe yR by proved been has lim →∞ 2 ytm twudb ovneti ipetasomto co transformation simple a if convenient be would it system, 1 pe bound upper 14)hsfudst ftresc rnfrain ht wh that, transformations such three of sets found has (1949)
/c k √ (0 = lim →∞ β y β # 2 1 = ”Teeedpit omthe form end–points These .” p . 4 ⇒ 1 = 1 = 1 = Γ , β Γ 2) EIBULL 4 1 i k − = 1 =Γ := URR hc sascae ihthe with associated is which aisfrom varies − − − − o h B the for Γ 4 EIBULL c Γ √ 3 exp exp exp  = 3 distribution  itiuin cuyargo hpdlk the like shaped region a occupy distributions − β + 1 Γ Γ ODRIGUEZ Γ + 1 1 ∞ (  1 Γ 3 2 For . 2 Γ 6 + − − − − − y ( uv nthe in curve k tecigfo h point the from stretching 2 URR k c k i c k Γ y 4 Γ Γ   √ ∞ → /c > c c ln 1 2 " − 1 2 1

.
2 β k y Γ 2 + 2 k  3 to Γ yeXIrgo.I h negative the In region. type–XII 1 c for / β , 3 + 1 2 1 (1977). − + ) . − 6 ): 2 URR eJ he fteB the of ∞ k 2 1 hr ilb utoemem- one just be will there , 1 3 .For ’. aro values of pair h gesml rnfrainis transformation simple ngle the rbudwt h W the with bound er Γ 3 oe bound lower y k ∞ → ,  c the OHNSON √  c 1 4 yeXIdsrbtosfor distributions type–XII y k cntn ie terminate lines –constant β c , c < c 1  cntn uv moves curve –constant . URR β , 2 oitcdistribution logistic + 2 3
. . . yeXIfamily. type–XII . pae compare –plane; ytm yanal- By system. 6 ftetype–XII the of #) the √ √ β c β –constant 1 points: 1 β , EIBULL and (3.4b) (3.4a) 2
103 uld β en ≈ 2 . . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC 0 ed othe to leads with ed othe to leads h est of density The h variate The nthe In where Taking Let den of families flexibility. three shape yields it variable, distributed mally ytmi endby defined is system h datg ihJ with advantage The 104 < Y < T a ω S easadrie omlvrae .. aigE having i.e., variate, normal standardized a be 2 L p 1 ytmtedniyi hto onra variate: lognormal a of that is density the system =exp(1 := := β β onra family lognormal et pntetransformation the upon rests Y S 2 1 f U ( slnal eae oavariate a to related linearly is Y ω y [ = = ytmwt none range unbounded with system = ) 2 nthe in ω − a f 4 /δ √ OHNSON 2 (  2 + y δ 2 2 cosh(4 2 1 S = ) ) π U ω B , ω p ( ytmi ie by given is system g 3 ω √ ( + 1 3 + stasomto sta,we netdadapidt nor a to applied and inverted when that, is transformation ’s Y S − δ 2 B 1 := L ln = ) π + ) 1) g y ω with /,A γ/δ, y 1 (  2 sinh = 2 Y T 1 a exp − exp / ln = ) 1 Y g 2 = > Y ( 3 cosh(2 A Y = 
( γ h − a , ln = ) Y − − 3 + X X := −  / 0 [ 2 +  1 h third The . γ g δ 1 λ 1 htw iht prxmt ndistribution: in approximate to wish we that − γ  Y ω B =4 := + : ω p − + Y ( ξ Y −∞ + ) cosh(2 Y ( δ 1+ (1 Y ω . 2 δ ) ln ω .  sinh )sinh(3 2) + a ( 2 2 < Y < 0 y T . Y ] ( ] iycre ihahg ereof degree high a with curves sity ω system 2 B . 0 = ) − 2  ) (2 1 + ) 1 2) + i > y , y B ∞  2 2 a , n Var and , S ) ) . , B B y , 0 0 sinh 3 + ) , ihbuddrange bounded with eae distributions Related 3 =3(2 3 := ( ∈ T 1 = ) R , ω B 1) + hnthe then ,  . (3.5b) (3.5d) (3.6b) (3.5c) (3.6a) (3.7a) (3.5e) (3.6c) (3.5a) - Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC n endby defined and trigfo h eue nfr variable uniform reduced the from starting them of J Some the families. further like of number great a exists There Miscellaneous 3.1.4 itiuini ebro h J the of member a is distribution h W The h einbuddby bounded (see region diagram the moment–ratio mome a the at for formulas Looking explicit and general no exist there and n ato I sfrsae ftedensity, the of shapes for As VI. of part and system, ie by given variate A oscasfiain hr n ls a pca rprya W property the to special respect a With has property. that class one me where two of classification, consisting families mous many have we least, not but Last te aiisaebsdo xasoseg,teG the e.g., expansions on based are families Other h est nthe in density The with W the and distributions of Systems 3.1 WORTH • • • • h xoeta aiyo distributions. of family exponential the and distributions ID distributions, stable distributions, location–scale f EIBULL ( S eisadteC the and series y S X OHNSON B = ) L . vrastpsI I I n ato I similarly VI; of part and III II, I, types overlaps eog othe to belongs S √ U uv scmltl oae nthe in located completely is curve δ 2 od o h orsodn einabove region corresponding the for holds π S ytm .. h T the e.g., system, B y ytmi ie by given is system (1 β X ORNISH 1 1 − 0 = := oainsaefamily location–scale y p ) h otmline bottom the , β β          F exp f 2 1 OHNSON X –F ( ln Y ( y ( x = ( = ISHER 1 = ) EIBULL  EIBULL λ | − − 1 ,b a, UKEY ω − Y ω γ (1 4 λ = ) S i.3/1 Fig. expansions. Y + − 2 + Y for − L distribution itiuinw ilpeetadinvestigate and present will we distribution  δ )( 1) slmd itiuin o variate a for distributions lambda ’s system. ihDF with Y F ω ln[ S β Y ) 0 3 U λ ω 2 S  y/ < y < n estha sees one ) fisCDF its if 3 + 2 − L 2) + suioa,but unimodal, is x (1 for for ein hstligta h W the that telling thus region, β RAM − b 1 ω − 2 a 2 − 1  y − λ λ –C 0 = 1 S )] 0 6= 0 = L 3
F br ny hsi dichoto- a is This only. mbers HARLIER S dteohrcasi missing is class other the nd nts. . nrlto oteP the to relation In . r ae ntransformations on based are 2 X U ) , . h system the t ( vrastpsI,V VII V, IV, types overlaps · n h onra curve, lognormal the and , ) a ewitnas written be may 0 S < y < B eis h E the series, a ebimodal. be may 1 S , B od for holds EARSON EIBULL X (3.7b) (3.7c) DGE (3.9) (3.8) and 105 - Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC xs constants exist tbei h ra sense broad the in stable Let c h titsense strict the a yeIeteevledsrbto ftemnmm(see minimum the of distribution value extreme type–I a has h o–rnfre variate log–transformed The hstasomto ilb fiprac hnmkn infer making when importance of be will transformation This where exponent itiuinas distribution h W The by fdsrbtosta r tbewith stable are that distributions of admvariable random A where needn dnial distributed identically independent uhthat such itiuinwith distribution a oainsaedsrbto.Ti stecs when case the is This distribution. location–scale a has n h CDF the and 106 xoeta itiuin(see distribution exponential e n cl aaee,respectively. parameter, scale and ter ingvnb 28 vdnl osntbln oti family, this to belong not does evidently (2.8) by given tion F Y (see 4 F ( ,X X, y aigdsrbtoswt fttis”tu niaigthe indicating thus tails,” “fat with rando distributions certain having model to used mainly are distributions Stable ( ) · 9 Chapters = ) EIBULL sterdcdCFwith CDF reduced the is d and eoe eult ndsrbto. qiaety denoti Equivalently, distribution.” in “equality denotes 1 α X , with F F 2 n . . . , X Y a itiuini o tbe ete ntebodsnenrin nor sense broad the in neither stable, not is distribution ( . 4 n · ( fteaoehlswith holds above the If . a 0 ) · ff). o vr tbedsrbto ehave we distribution stable every For epciey n has one respectively, , ) > 0 = eidpnetietclydsrbtdvrae.Tedist The variates. distributed identically independent be X α < osntdpn nayparameter. any on depend not does 0 scalled is F and : ≤ ( · F = ) fi sntcnetae toepitadi o each for if and point one at concentrated not is it if b 2 ( n X h aiyo G of family The . z et 3.2.1 Sect. 1 = ) ∈ F F nntl iiil distributed divisible infinitely = a a d ( n R ∗ x ( 0 = X · 1 = ) uhthat such ln = ) ( α − n, = b ∗ Y 1 2 = and exp F i)rno variables random iid) .Smtmsasu a Sometimes ). + Z =( := b n − ( b X =ln :=  C . · b n and exp ) n, X − 1 = ( ∗ 0 = X 2 AUCHY exp . . . AUSSIAN − n + 1 X h he–aaee W three–parameter The . b − + a o all for ∗ . . . ∗ osblt fa nnt variance. infinite an of possibility  )  1 = X  F cnmcpeoea seilyi finance, in especially phenomena, economic m z x b b + 2 n itiuin r tbewith stable are distributions −  stre the termed is b ( + /c. a X · X a ∗ c n =: ) a itiuin steuiu family unique the is distributions o n and tbetransformation itable . . . n,n ∗ hntedsrbto is distribution the then a w–aaee W two–parameter a has . =  unless S , + X neo h parameters the on ence b F n c.3.2.2 ect. gteCF of CDFs the ng ( r aldlcto parame- location called are n n, X 1 ∗ , = b /α ( 1 n · X , ) D)i o each for if IDD) ) c o oecharacteristic some for /a eue variable reduced eae distributions Related 3 1 = n, n ihCDF with ) 2 − X , . . . , .. ehv an have we i.e., ; h titsense. strict the EIBULL iuinof ribution b n a h same the has n X Z ∈ and tbein stable := n,n distribu- EIBULL N (3.10b) α (3.10a) n g b there X ∈ 1 = exist X ( and and X n, N is 1 ) . , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h xoeta aiypasa motn oei estimatin in role important an θ plays family exponential The Theorem: r etr fcmo dimension, common of vectors are where ucino h form the of function The unchanged. remain hstesaeprmtrcagsfrom changes parameter scale the Thus Then f gives Proof: hr h operator the where W the and distributions of Systems 3.1 u of sum a is IDD to related family. concept IDD A the of member a not is distribution sec ftetrsi h u.TeW The sum. the in terms the of each as umto,bti is it but summation, Theorem: variables. random 1 ( θ , . . . , x | xoeta family exponential θ S 1 θ 1 Inserting θ , . . . , S , . . . , saprmtr(both parameter a is n m Let Y i admvariables random iid eas ehv h following. the have we because Let ∼ F X Y m We i m ( F ) y X ∼ iid ∗ ( fa xoeta aiyadlet and family exponential an of r jointly are 1 = ) ( x 1 erdcietruhfraino h minimum the of formation through reproductive ,bn b a, eoe cnouin”Alsal itiuin r D.T IDD. are distributions stable All “convolution.” denotes We X , . . . , ) ie n(.3) notegnrlfrua(1.3a) formula general the into (2.33b), in given , 1 = 1 = f ( ,b c b, a, ( fcniuu itiuin scaatrzdb aigade a having by characterized is distributions continuous of − x 1 erdciiytruhsummation through reproductivity | /c − − − Q x θ sufficient c , = ) ( ) F and exp exp  θ n exp ) Y for X ) . ( ◦ i A   earno apeo size of sample random a be θ y m i  ( T EIBULL S 1 = ) ( − − a,o ore emultidimensional), be course, of may, i n h operator the and , θ 1 = j ( − n 1 = ) x  = B = ) and   b n b n , . . . , EIBULL y X ( i y to − n , . . . , y =1 x n distribution − − − − complete X exp ) j b b  b m 1 =1 1 T a  /c a a j √ −  c  ( Q  and X ) c n c  c itiuini o erdciethrough reproductive not is distribution F j    eog otesm itiuinfamily distribution same the to belongs ( i Q X n h oainadfr parameters form and location the and ) θ . Y ( ) ( n ⇒ θ for y T ◦ ) min( = ) j  eoe h inrpout”i.e., product,” “inner the denotes ◦ ( n We θ x T 1 ) θ , . . . , . ( n etn h parameters the testing and g x ,bn b a, ) X

enn htteCFof CDF the that meaning , , 1 X , . . . , − m n 1 /c when rmtedensity the from of c , n n
Q ) . i W iid then , m > n ( eW he θ ) and EIBULL EIBULL (3.11b) (3.11a) . T nsity 107 (  x  ) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC 31a ih(.2)oercgie htteepnnildi exponential the c that recognizes one (3.12d) with (3.12a) then distribution exponential reduced the have we and h ucin of functions the distributions: types three the parameter of location discovery a the Introducing on reported tions. we 1.1.1 Sect. In .. W 3.2.2 o,let Now, h rnfrain(.2)i eerdt sthe as to referred is (3.12b) transformation The ainhpads eaooiesol ehv ogte n d any forgotten have we should W the apologize involving O we so dynasties. distri European and the the lationship between between relationships relationships The family the edition. second its in ttsia itiuin xs nagetnme smyb s be may as number J great of a documentation in exist distributions Statistical W 3.2 parameter. scale the to respect with omparameter form esatwt ehp h otsml eainhpbtent between relationship simple most the perhaps with start We W 3.2.1 n te itiuin aeyteepnnildistributi exponential the namely distribution, other any xn n rmr fisprmtr r xoeta.We the When exponential. are bu parameters entirety, its its of more in or exponential one fixing not is family distribution a sR as h w–aaee xoeta itiuin( generalize (= DF distribution exponential two–parameter the notntl h he–aaee W three–parameter the Unfortunately 108 il fsfcetsaitc snthlfli euigW reducing in helpful not is statistics sufficient of ciple W 1 = f EIBULL ( OSIN x X fteW the of ) | ,b c b, a, a W a has R , EIBULL itiuini rcie(see practice in distribution EIBULL EIBULL AMMLER = ) EIBULL EIBULL EIBULL c 1 /1 Table f c b r ohkonadfie hnteW the then fixed and known both are ( OHNSON  y n xrm au distributions value extreme and n xoeta distributions exponential and | x itiuin n te aiirdistributions familiar other and distributions ,b a, n S and − distribution. itiuinwt DF with distribution b distribution. earv ttefloiggenera following the at arrive we , = ) a  /K PERLING c − 1 b OTZ 1 exp exp f ( Y 19,19,19)e l ossigo v volumes five of consisting al. et 1995) 1994, (1992,  y EIBULL  − = ) − = 1.2 Section 13)ddntigbtadn hr aaee to parameter third a adding but nothing did (1933) y a   − e ( b x X a − a − y b itiuini o xoeta;tu,teprin- the thus, exponential; not is distribution ∈ b > y , −  a ; oe–a transformation power–law R a  ( c a  y ) c  ercgiet recognize we ) 0 = n cl parameter scale a and ≥ , 0 ; , ,a a, EIBULL x b , n okn oteoiiso the of origins the to Looking on. EIBULL eti ufmle bandby obtained subfamilies certain t xoeta itiuin with distribution) exponential d ≥ tiuini pca ae(with case special a is stribution uin r smnfl sare as manifold as are butions 1) = ∈ eW he ,a a, emyesl vrokare- a overlook easily may ne e rmtecomprehensive the from een omlsfreteevalue extreme for formulas l oainparameter location R feteevledistribu- value extreme of itiuini exponential is distribution > b , srbtoa relationship, istributional apedt.Sometimes data. sample ihdensity with ∈ EIBULL R a W hat eae distributions Related 3 ,c> c b, , 0 . itiuinand distribution EIBULL b Comparing . ( > b 0 . a (3.12d) (3.12b) n the and (3.12c) (3.12a) swell as 0) into Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC yeI–aiu distribution Type–II–maximum h orsodn itiuin of distributions corresponding The yeIImxmmdistribution Type–III–maximum r hs ftemnmmadaegvnb h following: the by distribution given Type–I–minimum are and minimum the of those are yeImxmmdistribution Type–I–maximum W 3.2 5 nacuto t ucinlfr hsdsrbto sas s also is distribution this form bution functional its of account On EIBULL F f . F f II II M M III III M M itiuin n te aiirdistributions familiar other and distributions ( ( y y F ( ( f = ) = ) y y I I F M M f = ) = ) I I m m ( ( y y ( ( exp = ) = ) x x 1 = ) = )        c b      c b  exp  y exp 1 b a − b 1 ( b exp − b : −  : a  − exp 5 : y −  : − exp X   − X   Y exp c Y c  − y 0 − − − Y 1 ∼  a x − 1 1 y ∼ ∼ b a − ∼ −  b Ev − exp exp − b a b − Ev = Ev exp y Ev  a a i y −  −  ( ( II I − c − ,b a, c − III ( b  (  − ) − ( ,b a, Y exp x ,b c b, a, exp a   ( )  − − ,b c b, a, b for for ) for for a y or   a a − mtmscalled ometimes − − b x b ; ) )  X − y y a b ) y a < y a < y y   − ; ∼ b a ≥ ≥ − c   a c Lw a ) a  ,c> c a, < y ,      x , y ,        ( ,b a, y , . ; obyepnnildistri- exponential doubly ∈ ≥ ) ∈ R ,c> c a, ; R ; 0; 0; (3.17b) (3.13b) (3.15b) (3.14b) (3.13a) (3.17a) (3.15a) (3.14a) (3.16) 109 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC utbelgrtmctasomto.Satn rmteW the set t from transformed Starting be transformation. can logarithmic suitable distributions type–III and type–II The 3.3.2. Sect. in analyzed be will which efial eto hr rnfraino W a of transformation third a mention finally We hsi yeImnmmdsrbto hc a oainp location a has which distribution type–I–minimum a is This n hstasom(.9,)to (3.19a,b) transform thus and h rnfraingvni 31)masarflcinaota about reflection a means (3.16) in given transformation The X otetp–I–aiu itiuin(.5/)i the is (3.15a/b) distribution type–III–maximum the so ebro h xrm au ls.W aetefloigreci following the make We class. value extreme the of member yeIImnmmdistribution Type–III–minimum cl parameter scale yeI–iiu distribution Type–II–minimum 110 W W EIBULL EIBULL ∼ 6 ehv naaou eainhpbtentenra n h l the and normal the between relationship analogue an have We Ev iii F f F ( itiue variate distributed f distribution II ,b c b, a, II m m III III m M ( ( x x ( ( = ) = ) x x b ) ∗ = ) = ) F f or 1 = ( ( z z X 1 = ) = )        /c ; b c 6      c b ∼  hti h yeImnmmdsrbto scle a called is distribution type–I–minimum a why is That . twl eaaye nSc.3.3.4. Sect. in analyzed be will it 1  a We 1 − X x − − c b exp : − ( Z b − exp x exp ,b c b, a, :  a ln( = rW or exp ( −   X  c c − c − − Z − )  1 ( 1 ∼  1 EIBULL 0 X z − =  exp exp a Ev − x − exp − X b ln − (  ii b a x b − ( ) −  2 − b a ,b c b, a,  X , c ) distribution: − a    ( − c EIBULL c z .  ) x a exp − ≥ ) − − b b ln eetdW reflected for for for for a, x a  EIBULL gomldistributions. ognormal c b   ) c ( − ait,wihlast another to leads which variate,  yeIdsrbto sn a using distribution type–I a o z  c a < x x x a < x ) − . x , etclai at axis vertical ≥ ≥ arameter ,c> c a, < x , ln rcltasomto oa to transformation procal itiuin(.9,) we (3.19a,b), distribution a a b ≥ )      EIBULL         eae distributions Related 3 ,c> c a, . ; a ∗ 0; distribution ln = 0; y = b (3.21b) (3.19b) (3.18b) (3.19a) (3.18a) (3.21a) x (3.20) (3.22) n a and Log– = a , , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC itiuin,tegmafml,wihicue h W the includes W which family, the gamma of the distributions, transformation of kind some in The values. parameter distinct for .. W 3.2.3 ntels eto ehv ensvrlrltvso h W the of relatives several seen have we section last the In eolcto aaee.Tu yeI–aiu distribu type–II–maximum a Thus parameter. location zero a oprn 32ab ih(.4,)w e that see we (3.14a,b) with (3.23a,b) Comparing cl parameter scale d The am itiuin(.4) hnw nrdc hsscn s second the at this named arrive also introduce we we (3.24c), When version three–parameter (3.24b). distribution gamma pligtewl–nw ue ffidn h FadCFo tra a of CDF and DF the at finding arrive of rules well–known the Applying W 3.2 W S TACY sasaeparameter. shape a is EIBULL 8 7 f eue form reduced he–aaee am distribution gamma three–parameter ( ute eeaiain nrdcdb S by introduced generalization, further A c ugse edn o hsscin H section: this for reading Suggested 16) S (1962), x f ( EIBULL x ( | 16)itoue eodsaeparameter shape second a introduced (1965) x ,b ,d c, b, a, − | a EIBULL ,b d b, a, distribution eeaie am distribution gamma generalized ) d c TACY itiuin n te aiirdistributions familiar other and distributions − 1 b = ) = ) a ob hne against changed be to has /M no(.4)gvsthe gives (3.24a) into fagmadsrbto ihonly with distribution gamma a of f IHRAM n am distributions gamma and c ( ( x ( x f d x b twl eaaye nSc.3.3.3. Sect. in analyzed be will it , F | − f ( d c ,d b, 1 = − x ( ( (1965). z z a | Γ( a exp = ) = ) ) d = ) ) d eut nterdcdepnnildsrbto.Introduc distribution. exponential reduced the in results = ) d d c − ) 1 b − d x exp 1 x Γ( d − d b c exp − d 1 AGER   b ) 1 | TACY exp( d c − w–aaee am distribution gamma two–parameter Γ(  z b  exp( | Γ( . EIBULL  − x − − d /B ) d c − /M  − b  − AIN ) a nadtoa oainparameter location additional an has − z , b x/b 1 a x 8 IHRAN Z x   /A exp − − ) b a yeI–aiu itiuinwith distribution type–II–maximum a has c ) ait.Hr,w ilpeetacasof class a present will we Here, variate. 7 NTLE ; ;  orprmtrgmadistribution gamma four–parameter a ;   16) allows (1965), x x . − c x EIBULL n parameter one  ≥ c ≥ 17) P (1971), ≥  ( ; 0 ,a a, z > c EIBULL b 0; > d ,  x − inmyb aldan called be may tion ≥ ,d> d b, c itiuina pca case special a as distribution  ∈ 0) ARR a , R 0 < c ; aeprmtrit the into parameter hape itiuin originating distributions . notetwo–parameter the into ,d> d b, , a /W somdvral,we variable, nsformed 0 0 ∈ . EBSTER hrb h factor the whereby , a h DF: the has R ; ,c > d c, b, 0 . 16) S (1965), : a inverse : (3.24d) (3.23b) (3.24b) (3.23a) (3.24a) (3.24c) 0 n a ing TACY . 111 c in , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC n utsso he hc r yia o h omldistr normal of the values for different have typical we are measured which is three skewness of kurtosis a and fsens chosen: skewness of ehv enta hr xs auso h hp parameter shape the of values thes exist equating there in that chosen seen be have will we which criteria approximately the holds on only depends distribution normal the to bution h eainhp etoe ofraeeatweesterel the whereas exact are far so mentioned relationships The itiuint h W the to distribution .. W 3.2.4 ordering. stochastic a in resulting distributions these of h F(.4)cnan ubro aiirdsrbtosw distributions familiar of number a contains (3.24d) DF The 112 al 3/1: Table 3/1). Tab. (see values special L f f f f f f f f f f f IEBSCHER ( (    ( ( ( ( ( ( 9 x x x x x x x x • • • • x x x ugse o hsscin D section: this for Suggested | | | | | | | | | | c c c c | ,b, a, 0 c, b, a, 0 b, a, b, a, 0 d c, b, a, 0 0 0 , , , ≈ ≈ ≈ ≈ , , , √ 1 1 √ 2 √ c, , , , 3 3 3 3 2 2 1 1 EIBULL 1 2 1 2 . . . . pca ae ftegnrlzdgmadistribution gamma generalized the of cases Special , , ν , , , 25889 31247 43954 60235 , , , 16)cmae h w–aaee am itiuinadt and distribution gamma two–parameter the compares (1967) 1) 2 1) 1) 1) 1) 2 2 ν 2 , ) ) , , 1)  ν 2 2 1   for for for for EIBULL n omldistributions normal and iclrnra distribution normal circular distribution half–normal χ χ W three–parameter W reduced distribution exponential two–parameter distribution exponential reduced distribution the of Name R E dsrbto with –distribution 2 RLANG x µ µ α AYLEIGH dsrbto with –distribution ∗ 3 = = = 0 = itiuinwith distribution UBEY x x x ∗ 0 . , 0 .. o en=mode, = mean for i.e., , 5 distribution, . 5 .. o en=median, = mean for i.e., , EIBULL .. o oe=median. = mode for i.e., , 16a,M (1967a), distribution EIBULL distribution ν ν AKINO ere ffreedom, of degrees ν ere ffreedom, of degrees a ∈ 0 = N 9 (1984). distribution c n ie odtoso h parameters the on conditions gives and iigavleo eofrtemeasure the for zero of value a giving h ult fapproximation of quality The . c to fteW the of ation edn oasens fzero of skewness a to leading itiuin.I et 2.9.4 Sect. In distributions. e bto.Dpnigo how on Depending ibution. hen ν ν ,c b, ∈ ∈ eae distributions Related 3 N N and/or EIBULL elognormal he d r given are distri- Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ie by given hc sol eedn ntesaeparameter shape the on dependent only is which expression oteCFo h tnadzdW standardized the of CDF the to h ih-adsd f(.5)i h eirclo h coeffi the of reciprocal the is a (3.25c) of side right-hand The ewn oepot(.5)i oprsnwt 32a o the for (3.25a) with comparison in (3.25b) exploit to want We al 3/2: Table 3/2. Tab. α e 25a.Tesadrie omladW and normal standardized The (2.53a). see ntesqe ewl oka h lsns fteCFo h sta the of CDF the of closeness the at look will we median. sequel the the to In equal is mean the that such parameter aadrt E rate hazard eadn h kurtosis the Regarding W 3.2 M 5 3 3 3 3 2 4 0 = ...... AKINO 77278 60235 43954 31247 25889 25200 3 = c EIBULL und . 18)sget obs h prxmto ntema hazard mean the on approximation the base to suggests (1984) b Γ 0 0 0 0 0 0 ausof Values 1 = 1 ......  92573 90114 89892 89719 89645 88574 itiuin n te aiirdistributions familiar other and distributions h Γ + ( 1 . X τ T F p )  W = Γ 0 = p c, ( 2 α 0 0 0 0 0 0 τ X − Γ ...... 4 = ) Γ 18587 27787 28897 29834 30249 41619 . ehv w ausof values two have we , 90486 2 σ − 1 Γ , − 1 = 1 = 1 2 µ p a ob o–eaie i.e., non–negative, be to has Γ = Φ( 1 2 Γ Pr when 2 EIBULL − − τ X  − Γ = ) τ b exp exp X 1 − Γ − ≥ p  4 3 3 3 2 2 c 1 2 p σ −∞ − ...... Γ Z   98046 24306 11081 00730 96356 12819 a ≈ and τ − Γ − 2 µ variate p + 2 − 3 √   ≤ . − Γ Γ b 43927 Γ Γ 1 2 µ 2 Γ 1 Γ τ EIBULL 1 2 Γ 1  π 1 − 1 +  1 2 p +
c e c Γ τ σ = − , nodrt achieve to order In . hc snal h au fteshape the of value the nearly is which , ( Γ τ b 1 2 x c t µ 2 2 Remark q Γ Pr µ α α α ∗ . / ≈ − − i ≈ 2 4 3 4 ≈ ≈ itiuin aetesm mean same the have distributions Γ =Γ := d 2 X ≈ ≈ ≈ a Γ 2 x t . x x  25200 1 2 − 0 in fvrain(.8)when (2.88c) variation of cient 0 0 0 0 ∗ c ≤ . . 5 5   Γ µ 1 2 + 1 dr omldistribution normal ndard +  and i ausof values six c τ σ  c i ,  c
, ≈ F 5 aeE rate W . 77278 ( τ ) c  ≥ ( ie in given giving ) h (3.25b) (3.25a) (3.25c) ( 0 X the , 113 )  ; Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC nte eainhpbtenteW the between relationship Another itiuin ilb icse ncneto ihteF 3.3.7.2. the and 3.3.7.1 with Sections connection in discussed be will distribution, τ a.33as ie h differences the gives also 3/3 Tab. ie by given Using 114 = • • • − h orcsso orsodn kweshv ifrn per different have skewness corresponding of cases four The dist both of smaller. zero) much (= skewness corresponding to respect With qiaec ftekroi eas hslast h maximu the to leads this because kurtosis the of W equivalence a base to advisable not is It a.3/2 Tab. 3 ∆ . (0 0 (1) – – – omldsrbto a eapoiae aifcoiyb satisfactorily approximated lo be the can distribution with normal associated a probabilities say, may we Generally, itiuinwt h hp parameter shape the with distribution boueerrfor error absolute τ − − other. ouedifference solute oeo h orvle of values four the of None 0 c for efidthat find We ( . τ . )3 1) 3 = 0105 1 1 = ) . .

h i W six the 5 8 F F F F F F 0 = − , . . ≤ W W W W W W 43945 0 (6) (5) (4) (3) (2) (1) 1 − c for oka a.33rvastefloigfacts: following the reveals 3/3 Tab. at look A . , τ . 1 ( ( ( ( ( ( 6 0355 ≈ τ τ τ τ τ τ . τ , − ≤ 0 c 1 = ) 1 = ) 1 = ) 1 = ) 1 = ) 1 = ) c , 3 = EIBULL 3 ≤ 3 = max 0 = . for 1 60235 τ . . τ 1 25889 τ . , .

c 60235 1 ∆

− ≤ = 0 ∆ ≈ − − − − − − and . Ds hc ilb oprdwith compared be will which CDFs, (5) 2 (3) ed otesals boueerrfor error absolute smallest the to leads − 2 exp exp exp exp exp exp . EIBULL ∆ ≤ ( . 1 0 EIBULL 52200 ( τ τ atce to (attached . τ . ( ) 7 1 τ i )

) 1 =       c τ ,

( − − − − − − ≤ and 0 = τ ihzr–kwesi nfrl etrta any than better uniformly is zero–skewness with 0 = (0 (0 (0 (0 (0 (0 = ) 0 . 0 = and prxmto ftenra itiuino the on distribution normal the of approximation 9 ...... n omldsrbtos ae pnthe upon based distributions, normal and 27 0 + 92573 01 0 + 90114 99 0 + 89892 91 0 + 89719 94 0 + 89645 87 0 + 88574 9 n,finally, and, ; 1 0079 . 0099 . and 0 . F

0 ∆ W α ( c ≤ and i
(6) ) 3 ≈ 2 ( for olwdby followed . τ 0 = τ 1 ( 3 ) 1 τ AUCHON . ≤ . c ≤ ) ...... − 03 ( 60235 7 18587 27787 28897 29834 30249 41619 ilstesals aiu ab- maximum smallest the yields )

3 = τ ≤ 0 = Φ( 1 c ≤ . τ 6; ≈ . τ 31247 . ) 3 ≤ 0274 τ τ τ τ τ τ c 3 . for ) ) ) ) ) ) ta.(96 xesosin extensions (1976) al. et 0 . c max 1 5 3 3 3 3 2 ≈ 43954 ; roso l i cases: six all of errors m ...... formances. 77278 60235 43954 31247 25889 25200 τ iuin,teerr are errors the ributions, c 8 ≈ i 3 for . and ≈ eae distributions Related 3 . Φ( 1 = 25889) sn W a using y

3 ∆ e upr alof tail (upper) wer 3 c       . ie h smallest the gives 25889 τ . max (4) . , , , , , 31247 ≈ ) , τ Tb 3/3 Tab. in ( − 2 5 τ . , . . . , . 3 )

77278 ∆ .

0 sbs for best is sbs for best is 0 = (2) ≤ EIBULL ( . τ 6 . 0088 τ ) are , χ and

2 ≤ = – Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC al 3/3: Table uigteseilsaefactor scale special the tuting hc sesl eonzda h est fa of density the as recognized easily is which 32a ie W a gives (3.26a) ntecneto h eeaie am itiuinw ment we distribution the is gamma them generalized of the of context the In W 3.2.5 W 3.2 W − − − − − − − − − − − − − − − − − τ 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 EIBULL ...... 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 ...... Φ( 0808 0668 0548 0446 0359 0287 0228 0179 0139 0107 0082 0062 0047 0035 0026 0019 0013 EIBULL f τ ( ) x F density: | EIBULL ...... W 0000 0657 0476 0325 0204 0112 0050 0014 (1) ν ausof Values − − − − − − − − − − − − − − − − − − = ) itiuin n te aiirdistributions familiar other and distributions ( χ τ –distribution ∆ ) − − − − − − − − √ EIBULL (1) ...... 0178 0150 0192 0223 0242 0247 0237 0214 Γ 2 n ute distributions further and ( Φ( τ 2  ) F τ f ν 2 ...... W 0835 0679 0543 0426 0327 0245 0178 0125 0084 0053 0031 0017 0008 0003 0001 0000 ) (2) itiuinwith distribution (  − − F , x ( τ  | √ with 0 ∆ ) W ( √ 2 i b, , − − − − − − − − − − − − − − ) x . . ( 0027 0010 yagnrlsaeparameter scale general a by (2) 2 ...... τ 0005 0020 0032 0042 0049 0054 0055 0054 0051 0046 0039 0032 0025 0019 )= 2)  ) ν ( ν τ and − ere ffedmwihhsteDF: the has which freedom of degrees ) 1 F ...... W 0839 0684 0549 0432 0334 0252 0185 0131 0089 0058 0035 0019 0009 0004 0001 0000 0000 (3) exp 2 b ∆  ( ( a τ i x ( b ) ∆ ) R ( 0 =  − − − − − − − − − − − − − − − τ . . . AYLEIGH 0031 0016 0001 (3) ) ...... exp 0013 0025 0035 0043 0048 0050 0050 0047 0043 0037 0031 0025 0019 0013  for b , ( √ τ x )  2 i F = − ......  1 = W 0848 0695 0562 0446 0348 0266 0199 0144 0101 0068 0043 0026 0014 0007 0003 0001 0000 (4) 2 √  ) ( distribution τ x b 2 , x , ∆ )  2 and − − − − − − − − − − − − − 2 , . . . , . . . .  oe t pca ae.One cases. special its ioned 0041 0027 0014 0001 > b (4) ...... 0011 0021 0029 0035 0038 0040 0039 0036 0033 0028 0023 0018 0013 ≥ , ( c τ 0 ) 0 2 = ν , F 6 ed otefollowing the to leads ...... W 0858 0708 0576 0462 0365 0283 0215 0159 0115 0080 0053 0034 0020 0011 0005 0002 0001 (5) . for ∈ ( τ ∆ ) N ν − − − − − − − − − − − − ...... 0051 0040 0028 0017 0006 (5) ...... 2 = 0004 0013 0019 0024 0027 0028 0028 0026 0024 0020 0017 0013 ( τ ) Substi- . F (3.26b) (3.26a) ...... W 0909 0777 0661 0559 0470 0392 0325 0268 0219 0178 0143 0114 0090 0070 0054 0041 0031 (6) ( 115 τ ∆ ) ...... 0101 0109 0113 0113 0110 0105 0098 0089 0080 0070 0061 0052 0043 0036 0029 0023 0018 (6) ( τ ) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 al 3/3: Table − − − − − − − − − − − − − © 2009 byTaylor& FrancisGroup, LLC 116 1 1 1 0 0 0 0 0 0 0 0 0 0 1 τ 0 0 0 0 0 0 0 0 0 1 1 1 1 ...... 2 1 0 9 8 7 6 5 4 3 2 1 0 3 ...... 1 2 3 4 5 6 7 8 9 0 1 2 3 ...... Φ( 9032 8849 8643 8413 8159 7881 7580 7257 6915 6554 6179 5793 5398 5000 4602 4207 3821 3446 3085 2743 2420 2119 1841 1587 1357 1151 0968 τ ) F ausof Values ...... W 7358 7059 6742 6409 6060 5699 5328 4948 4563 4175 3788 3405 3030 2665 2314 1980 1666 1374 1108 0868 8921 8754 8570 8366 8143 7901 7639 (1) ( τ ∆ ) − − − − − − − ...... 0101 0144 0188 0229 0268 0301 0328 0346 0355 0354 0342 0320 0287 0245 0195 0139 0079 0018 0020 0059 (1) ...... 0043 0100 0111 0095 0074 0047 0016 Φ( ( τ ) τ F ) ...... F , W 8989 8800 8590 8357 8102 7826 7528 7212 6878 6528 6167 5795 5417 5036 4654 4275 3902 3538 3185 2847 2525 2221 1937 1673 1431 1210 1012 (2) ( W τ ( i ∆ ) ) ( − − − − − − − − − − − τ ...... 0003 0019 0036 0052 0067 0081 0092 0100 0104 0105 0103 0096 0087 0074 0060 0044 (2) ...... ) 0043 0049 0053 0056 0057 0056 0052 0046 0037 0026 0012 and ( τ ) F ∆ ...... W 3177 2840 2519 2217 1934 1672 1431 1212 1015 8992 8803 8592 8358 8102 7824 7525 7207 6871 6521 6158 5786 5407 5025 4643 4264 3891 3528 (3) ( i ( ) τ ( τ ∆ ) ) − − − − − − − − − − − − ...... 0091 0097 0099 0098 0093 0085 0075 0062 0047 0009 0025 0041 0057 0071 0082 for (3) ...... 0040 0046 0051 0055 0057 0058 0056 0051 0043 0033 0021 0007 ( τ i ) 1 = F ...... W 3157 2823 2506 2207 1927 1669 1432 1216 1022 9001 8810 8597 8360 8101 7819 7517 7196 6857 6503 6138 5763 5383 5000 4618 4239 3868 3506 (4) , ( τ 2 ∆ ) , . . . , − − − − − − − − − − − − − − ...... 0072 0080 0086 0088 0087 0082 0075 0065 0054 0016 0032 0047 0060 (4) ...... 0031 0039 0047 0053 0059 0062 0063 0062 0058 0051 0041 0029 0015 0000 ( τ 6 ) F eae distributions Related 3 (Continuation) ...... W 3134 2803 2489 2194 1919 1665 1432 1220 1029 9012 8819 8603 8363 8100 7815 7508 7183 6840 6483 6114 5737 5355 4971 4588 4210 3840 3480 (5) ( τ ∆ ) − − − − − − − − − − − − − − − ...... 0049 0060 0070 0076 0079 0078 0075 0069 0061 0003 0019 0035 (5) ...... 0020 0030 0040 0050 0059 0067 0072 0075 0074 0071 0065 0055 0043 0029 0014 ( τ ) F ...... W 2937 2631 2345 2079 1835 1611 1407 1223 1057 9131 8921 8683 8415 8119 7799 7455 7092 6713 6323 5925 5524 5125 4730 4343 3968 3608 3263 (6) ( τ ∆ ) − − − − − − − − − − − − − − − − − − − ...... 0024 0050 0072 0089 0099 0072 0039 0001 (6) ...... 0148 0112 0075 0039 0006 0040 0083 0125 0166 0201 0231 0254 0268 0274 0270 0258 0239 0213 0182 ( τ ) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC pc,is space, hti,tedsac rmteoii oapoint a to origin the from distance the is, that al 3/3: Table W 3.2 xuss R Excursus: hsdsrbto a nrdcdb ..S J.W. by introduced was distribution This E faosis Let acoustics. of 1 1 1 1 3 2 2 2 2 2 2 2 2 2 2 1 1 τ ...... ( 7 6 5 4 0 9 8 7 6 5 4 3 2 1 0 9 8 X ...... Φ( 9987 9981 9974 9965 9953 9938 9918 9893 9861 9821 9772 9713 9641 9554 9452 9332 9192 i 0 = ) τ EIBULL ) F ...... W 9960 9949 9935 9919 9899 9874 9845 9810 9768 9719 9661 9593 9515 9424 9321 9203 9070 (1) ∀ ausof Values i ( AYLEIGH τ n Var and itiuin n te aiirdistributions familiar other and distributions ∆ ) f − − − − − − − − − − − − − − − − − X ( (1) y ...... 0027 0033 0039 0046 0055 0064 0073 0083 0093 0103 0112 0120 0126 0130 0131 0129 0122 1 = ) X , ( ( τ Φ( X ) distribution (2 2 F i τ = ) X , . . . , ...... W 9989 9985 9978 9969 9957 9941 9921 9895 9861 9819 9767 9704 9627 9535 9427 9301 9155 (2) ) σ F , 2 ( ) σ τ n/ ∆ ) W 2 ( i 2 − − − − − − − − 2 ∀ ) ...... ( 0003 0003 0004 0004 0004 0004 0003 0002 0000 Γ( (2) ...... n i τ 0002 0005 0009 0014 0019 0025 0031 0037 h est ucinof function density The . ) ( n/ ea i apeo size of sample iid an be τ and ) 2) Y F ...... W 9865 9823 9772 9708 9632 9540 9432 9306 9159 9990 9985 9979 9971 9959 9944 9924 9898 (3) TRUTT y = ∆ n ( − ( t u u v τ ( i 1 ) X ∆ ) ( X i exp − − − − − − − τ =1 1 n ...... X , . . . , Lr R (Lord 0004 0002 0003 0004 0005 0005 0006 0006 0006 0005 ) (3) ...... 0001 0004 0009 0014 0020 0026 0033 X for  ( τ i 2 − ) , i 2 F y 1 = ...... W σ 9873 9832 9781 9719 9643 9552 9444 9317 9169 9991 9987 9982 9974 9963 9949 9929 9905 (4) AYLEIGH 2 n 2 ) (  τ , nthe in n ∆ ) ; 2 − − − − rmanra itiuinwith distribution normal a from , . . . , ...... > y 0012 0011 0009 0006 0002 0005 0006 0007 0008 0010 0011 0011 0012 (4) . . . . 0003 0008 0015 0023 14 99 naproblem a in 1919) – (1842 ) n ( τ 0 dmninlE –dimensional ) > σ , 6 F ...... W 9882 9843 9793 9732 9657 9566 9458 9330 9182 9993 9990 9985 9977 9968 9954 9936 9913 (5) (Continuation) ( τ 0 ∆ ) . − − ...... 0021 0021 0021 0019 0016 0012 0006 0007 0008 0010 0012 0014 0016 0018 0020 (5) . . 0002 0010 ( τ UCLIDEAN ) F ...... W 9950 9924 9889 9840 9777 9695 9592 9465 9312 9999 9999 9998 9996 9993 9988 9980 9968 (6) (3.27) ( 117 τ ∆ ) ...... 0089 0103 0116 0127 0136 0140 0140 0133 0120 0013 0017 0023 0031 0039 0050 0062 0075 (6) ( τ ) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC F ihteCFo eue W reduced a of CDF the with i.3/3 Fig. For ihtelgsi n omlCDFs. normal and logistic the with 32b saW a is (3.29b) ceietetria tt frpiae.H trsfo th from starts He CDF: the replicates. for of state terminal the acterize hsi ierhzr ae hs h W the Thus, rate. hazard linear a is This ls of class hsclpoessta nov ulainadgot rre or G growth by analyzed and nucleation involve that processes Physical D fterdcdlgsi distribution logistic reduced the of CDF distribution logistic h omladtelgsi itiuin r eysimilar, kurtosis very more are has distributions distribution logistic logistic the and normal The h fet of effects The For h aadrt eogn oteseilW special the to belonging rate hazard The With 118 ae(3.26b). have W (5) • • • > n n ( n τ nraigtevleof value the Increasing nraigtevleof value the Increasing nraigtevleof value the Increasing 1 = ) = oyoilhzr aedistributions rate hazard polynomial hw h etfitn W fitting best the shows Tb 3/3 Tab. in 1 ν h ouinis solution the h ouinis solution the and ,n m, EIBULL ITTUS σ d 1 = F d ( x ie os eut hnteapoiaint the to approximation the than results worse gives F and x qain(.7 sthe is (3.27) equation , yasmercvrino h W the of version symmetric a by 16)t riea h ls fdsrbto ucin that functions distribution of class the at arrive to (1967) ) ( itiuinwith distribution x = K ) m,n x K F ntefr fteCF ie y(.9) r sfollows: as are (3.29c), by given CDF, the of form the on ( F n m K x 1 = m h ) ( eue t height. its reduces m, ( τ nrae h tens ftecnrlpr ftecurve. the of part central the of steepness the increases EIBULL ae h uv oeupright. more curve the makes EIBULL  x = ) 1 1 − | − 1 = 0  b, , F 1 ( ( a n EIBULL − α ( − )= 2) CDF m x itiuini h ymti cases symmetric the in distribution χ 4 0 = − exp ) dsrbto 32a,adwith and (3.26a), –distribution exp EIBULL  1 + 4 = 1) n b , ; ( 2 b  K h oyoilbigo ereone. degree of being polynomial the , c . > K itiuinwith distribution − − 1 x 2) b ≈ = x K τ/ π tsesovost prxmt the approximate to obvious seems It . = itiuin(.6)is (3.26b) distribution m 3  . ( +1 60235 m 0 m x EIBULL b 2 > m , 2 m √ 1 + 1 + 1) + x. +1 3 olwn ifrnilequation differential following e
aainpeoeahv been have phenomena laxation ohbigsmerc u the but symmetric, being both   giving 1 /K / − itiuin oprn the Comparing distribution. . (1 1  − c n , 1 n / 2 = α ) ( m 3 eae distributions Related 3 . ≥ omldistribution. normal +1) 0 = n samme fthe of member a is 1 2 = . and ncomparison in ) and hudchar- should c F = W σ (2) (3.29b) (3.29c) (3.29a) = m ( (3.28) τ b 1 + ) we , to . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC enie hslaigt ieiecutdi aua unit natural in still counted is lifetime it geometric a whether month to a leading or revol thus week or meantime, a shocks hour, cycles, an runs, once inspected blows, measu is loads, data work failure of encounter number often as we fields these In theory. o h xoeta n am itiuin,respectively distributions, gamma and exponential the for . oictoso h W the of Modifications 3.3 ra ubro itiuin aebe eeoe httak that developed been have distributions of number great A anae fapiainfrteW the for application of area main A W the of grouping generaliza the other that hope while mo We practice these science. of from Some questions departure. to of response point their as (2.8) tribution .. iceeW Discrete 3.3.1 incomplete. nor overlapping neither is tions iue3/3: Figure W the of Modifications 3.3 11 10 aaee siainfrdsrt W discrete for estimation Parameter ugse edn o hsscin M section: this for reading Suggested A BOUAMMOH n the and Dso h tnadzdlgsi,nra n W and normal logistic, standardized the of CDFs (1989). eaiebnma distributions binomial negative EIBULL EIBULL distribution distribution URTHY EIBULL EIBULL EIBULL /X IE EIBULL 11 itiuini ieiersac n reliability and research lifetime is distribution itiuin stetdb A by treated is distributions /J IANG distribution (2004). fsot ntefloigtnsubsec- ten following the in offshoots r nw ob iceealternatives discrete be to known are eaeitrse,fo the from interested, are We . in aeteroii npure in origin their have tions ftm.I hscnet the context, this In time. of s ictoshv enmd in made been have difications e sdsrt aibe such variables discrete as red h lsia W classical the e EIBULL tos oeie device a Sometimes utions. okn rhsfie nthe in failed has or working 10 distributions LI K HAN EIBULL /K HALIQUE dis- 119 / Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC edfieadrdfietefloigconcepts: following the redefine and variable define We lifetime discrete a for Functions Excursus: distr discrete what practice, and theory both of viewpoints 120 hrceitco h otnosW continuous the of characteristic h otfeunl sdadwd–peddfiiino h ha the of definition wide–spread 1 and fotne used frequently most The The u rto l ehv ocmeto h ucin htgenera that functions the on comment to have we variable. re lifetime mean all and of rate hazard first of But notions the generalize to tries but ean h omo h otnosCF The CDF. continuous the of form the retains Ro h otnosvrini h es htisCFadisH its and CDF its that sense the W in continuous version continuous the of HR eso.Fnly emninaohrapoc,ta fP of that approach, o another each mention to them we compare Finally, and types version. two these present first will We W discrete a find to impossible is ugse yS by suggested W S ALVIA EIBULL • • • yeIdsrt W discrete type–I alr itiuinCDF distribution failure rbblt ffiuewti the within failure of probability uvvlfnto CCDF function survival with with 19) hsapoc osntsatfo h otnosW continuous the from start not does approach This (1996). nCatr2 is 2, Chapter in itiuin h nwrt hsqeto sntuiu,dep unique, not is question this to answer The distribution. EIBULL TEIN /D h F itiuinfritga auso h variate. the of values integral for distribution R ATTERO k EIBULL ( ( ∞ ∞ F := = = = 1 = ) 0 = ) ( k R := ) Pr P Pr F R Pr ( 18) ean h omo h otnoshzr ae It rate. hazard continuous the of form the retains (1984), F , R , distribution k k ( ( ( 1 ( ( := ) EIBULL k X k X X := EIBULL Pr − k R ) − ( ( t nto ie( time of unit –th ( − − − = ( ≥ = F = Pr X k 1) ): 0 := 1) ): 1 := 1) ( F k k k ( − k ≤ X − ) ) ( | − Pr i k X 1) = itiuini ob preserved. be to is distribution k X R itiuinta iisbt h D n the and CDF the both mimics that distribution = ∞ − ( k = ) 1) k > X ≥ +1 ( nrdcdb N by introduced , k k 1) and and ); ) k yeI iceeW discrete type–II P X i ); =0 i k ; k P 1 = ) P k P 0 = rbblt asfunction mass probability k k k i 0 = 0 = ; = = , k 1 − F R , , , ADGETT 1 0 = bto ih orsodt the to correspond might ibution 1 ( ( 2 adrt ntedsrt ae see case, discrete the in rate zard F , k k , . . . , iullf otedsrt case. discrete the to life sidual 2 2 ) ( − , . . . , , k . . . , − 1 ) AKAGAWA 1) , ; hradt h continuous the to and ther F ge ihtoeo the of those with agree R 2 . . . , − ( /S                            k l ecieadiscrete a describe lly EIBULL R PURRIER eae distributions Related 3 − ( EIBULL k 1); ) . nigo what on ending /O SAKI distribution ) distribution 18)and (1985) (1975), (3.30b) (3.30d) (3.31a) (3.30a) (3.30c) (3.30e) , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ihtecrepnigcmltv function cumulative corresponding the With earv at arrive we ewl term will We aadrt function: rate hazard otnoscs,wihaet efudin found be to are which case, continuous especially nta ehv 33c.Frti esnR reason this For (3.32c). have we instead h eainhp between relationships The entc h olwn rpriso (3.31a): of properties following the notice We rate hazard cumulative corresponding with W the of Modifications 3.3 12 • • • • o iceehzr ucin,see functions, hazard discrete For h enE mean The hs odtoa rbblte n h nodtoa pro unconditional the and probabilities conditional These h uvvlfunction survival The h k sacniinlpoaiiy thus probability, conditional a is λ k the ( X suohzr function pseudo–hazard ) fi xss sgvnby given is exists, it if , R ( k h (1 = ) P P EIBULL R R ( 0 k λ Λ( x ( ( k k ) k H , ) k =ln := = = ) E := ) exp 6= − n h aadrate hazard the and ( X distribution S ( h h h x ALVI = ln = R  = ) 0 0 k ) (1 ) ( R (1  k nteoesd and side one the on H exp = ) − OY ( R − i X i P 2/1 Tab. − k =0 ( =1 /B k − k k ( H ln R − /G k h := ) 12 OLINGER λ h 0 R ) ( oa odfeetaei rmtehzr rate hazard the from it differentiate to as so ( k R i 1) 1 k − UPTA − ≤ ( ) ) ( k 1 1)    k · X = ) ) i h onthold not do , =0 ) . . . − exp = k ; · k , − . . . 19)hv rpsda lentv discrete alternative an proposed have (1992) h Λ( ≤ k h e (3.30e) see · ln k X k ∞ i (1982). =1 (1 0 = k . · 1 r ikdas linked are R ) (1 " . − j  Y ( − =0 . k k , − F h 1 ) X i (1 k , =0 k ( h babilities 2 ); x with 0 − . . . , ) ); h R ,            h i h # j k 0 = , ; h ) ( . k ≥ x ) and , P 1 1 nteohrsd nthe in side other the on k . , 2 r ikda follows: as linked are    H . . . , ( k ) endabove, defined h (3.33b) (3.32d) (3.32b) (3.31b) (3.33a) (3.33c) (3.32c) (3.32a) k . 121 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h greater ahohrbut other each 33f ntelwrpr for part lower the in (3.34f) 3/4 Fig. n t eairi epneto response in behavior its and hs h suohzr ucinflosas follows function pseudo–hazard the Thus, n h CFby CCDF the and o h parameter the So, h aadrate hazard The iisteCFo h otnosW continuous the of CDF the mimics asfunction mass The 122 n h aadrt codn o(.1)as (3.31a) to according rate hazard the and h rbblt asfnto olw as follows function mass probability The k I sicesn u scnaefor concave is but increasing is • • • yeIdsrt W discrete type–I Pr sicesn for increasing is sdcesn for decreasing is a h osatvalue constant the has β ( hw h aadrt 33c nteuprpr n th and part upper the in (3.34c) rate hazard the shows . X λ k I ≥ nrae ieryfor linearly increases λ P k k I k = ) I ( ,β q, β ( k ly h aerl as role same the plays X j ∞ = ) 0 = > β λ k 0 h > EIBULL k I P < β < F h P , j = I k I 1 k I I k I 1 R , ( = 1 = 1 ( q h . 2 = k ,β q, I k . . . , − 1 = ) β q ( 0 = β q k k 1 q − distribution k stesm sta of that as same the is β = ) − ) β ) ; h ifrnebigtegetrtesmaller the greater the being difference the , and β for β . q ( − 9 nietydfie by defined indirectly ( − k k 2 = 2 = k q and β q +1) 1) + ( 0 = q EIBULL k ( k ( 1 = +1) . hc ssmlrt h otnoscs,whereas case, continuous the to similar is which , k +1) β +1) β , c − β β 1 k , ntecniuu ae h D sgvnby given is CDF The case. continuous the in β i 0 = , ; β β ; 2 ln nrdcdb N by introduced ; ; . . . , k itiuin hycnie probability a consider They distribution. k q k . k 0 = 5 ; 0 = , 0 = 0 = 0 ; k 1 , . , 0 = 1 h 0 , , 1 < q < , k I , 1 1 , 2 . , , 2 1 . . . , , 2 2 . . . , . 1 , . . . , . . . , 5 , . 2 1 . . . , h AKAGAWA ; suohzr function pseudo–hazard e ; k I and and eae distributions Related 3 > β λ k I /O r o qa to equal not are 0 . SAKI q n/rthe and/or (1975) (3.34d) (3.34b) (3.34e) (3.34a) (3.34c) (3.34f) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC where Remark: eseta 33d n 33)hv h aedul exponenti double same the have (3.35) all and for (3.34d) that see we iceeW discrete hn h rnfre variate transformed the Then, n hence and and oprdwt h D ftetoprmtrcniuu W continuous two–parameter the of CDF the with Compared iue3/4: Figure W the of Modifications 3.3 x b 1 = 1 = ups htadsrt variate discrete a that Suppose X EIBULL k aadrt n suohzr ucinfrtetp- discr type-I the for function pseudo–hazard and rate Hazard tribution Pr ( 1 + /b a h iceeW discrete the has ( c Y stecmie cl–hp atro (2.26b). of factor scale–shape combined the is = k itiuinrdcst h emti itiuin hst This distribution. geometric the to reduces distribution k 0 = = ) F β ( , EIBULL x = 1 q p , | Pr c 2 0 k . . . , ,c b, , X ( − and X 1 EIBULL distribution = ; ) ≥ 1 = ) if Y k q Pr k 1 exp( = 1 = = ) /β ( Y − > β , itiuinitoue bv.When above. introduced distribution Y , ≥ Pr 2 exp a emti itiuin i.e., distribution, geometric a has . . . , ( k − Y = ) n 1 0 /b ≥ ; − ilhave will , p c q  k exp( = ) k + β x b . = )  q c o 1 = q x , k β − EIBULL , and b ≥ 1 lfr n hycoincide they and form al ) 0 , 0 , < p < distribution t W ete 1; ransformation EIBULL β 1 = (3.35) the , dis- 123 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC is i.e., 33)ad(.7)cicd at coincide (3.37a) and (3.36) hc sncsayt ensure to necessary is which m ftecniuu W continuous the of iisteHR the mimics The q n otnosdsrbtos epciey when respectively, distributions, continuous and hs uhr n K and authors These aen lsdfr nltclepesos hyhv obe to have they expressions; analytical closed–form no have stecutratt h oe–a eainhplnigth linking relationship power–law the to counterpart the is 124 h oet ftetp– iceeW discrete type–I the of moments The K W f(3.34b). of HAN EIBULL satucto au,gvnby given value, truncation a is α yeI iceeW discrete type–II sacmie cl–hp atr h rbblt asfunc mass probability The factor. scale–shape combined a is ta.(99 ietefloigieult o h means the for inequality following the give (1989) al. et h distributions. k II = P    k II EIBULL k α ULASEKERA = = 0 β m − EIBULL k α    1 = E itiuinby distribution h ( h h β for for x    k II X − k II 1 II = h 1 r ≤ int (1 ( k = ) j β Y 19)so o oetmt h w parameters two the estimate to how show (1994) x k =1 − k k µ distribution 1 = − 1  | for d e (3.32a). see ; 0 = 1 = (1 α 0 X k EIBULL − h ,c b, , c ∞ − =1 ∞ − k II 1 , 1 and − / 2 k or ( j α 1 µ < = ) β m , . . . , r ) −  . . . m > k β 1) α q distribution − c k  b c = q 1 (1 µ < β nrdcdb S by introduced );  exp( = − c/b if if − x b d k q     h . ( c c β > β k 1 = − 1 II , +1) > α , 1 ≤ ) xoeta n h continuous the and exponential e − , β 1 1 1 2  for /b m. , . . . , vlae ueial.A numerically. evaluated    0 c > β , ) , µ : ingnrtdb (3.37a) by generated tion k TEIN d ≥ and eae distributions Related 3 2 0 /D , . µ ATTERO c ftediscrete the of (3.37b) (3.37c) (3.37a) (1984) (3.36) β and LI Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC .. eetdaddul W double and Reflected 3.3.2 (3.37a). to similar is which variate eciemn aadrtso iceedsrbtos h p The distributions. is (3.38a) discrete to of sponding rates hazard many describe o utbevle of values suitable For h oictoso hsadnx w etoscnito som of consist sections two W continuous next a and of this of modifications The n h uvvlfnto is function survival the and h rbblt asfnto is function mass probability The where 33d ncmiainwt 33bc ie h pseudo–haz the gives (3.33b,c) with combination in (3.37d) n aaee.Tedsrt aadrt fti oe is model this of o rate increasing hazard constant, be discrete The may parameter. rate one hazard its that fact the is For h orsodn uvvlo eiblt ucinis function reliability or survival corresponding The W the of Modifications 3.3 h iceedsrbto ugse yP by suggested distribution discrete The distribution. type–I the does as o iia nfntoa omt n ftefntosdescri functions the of any to form functional in similar not itiuin h nyie oncigti itiuint distribution this connecting item only The distribution. C OHEN • • • β eraigfor decreasing nraigfor increasing osatwith constant 1 = h X k III 17) rgntsi h olwn iertransformation linear following the in originates (1973), R h : P k III II k h yeI iceeW discrete type–II the III is ( k 1 = = ) exp = − EIBULL j Y 1 =1 k > β < β exp R h − > d − III 1 exp( d h EIBULL 0 0 − − , λ ( ( . ait.The variate. k k k III d 0 h exp = ) − Y 1) + ( k II h parameter the k d λ =
− ) 1) + k II distribution = for EIBULL β d a = i EIBULL ( j Y = k   j Y =1 β β k =1 − k i − 1) + − 0 = ADGETT  ; ln eetdW reflected d exp ( 1 k X X k j itiuinrdcst emti distribution geometric a to reduces distribution  =1 − +1 , β 0 = 1 β − ;  distributions j α − j − nterange the in k a β , /S j d k α = )   1 β 0 = , − PURRIER ; 2 β 1 β . . . ,   a EIBULL − , k 1 ; 1 ; − 0 = , h otnosW continuous the o  2 k ; k X. . . . . , eraigdpnigo only on depending decreasing r > d 0 = r function ard 1 = , igacniuu W continuous a bing ed–aadfnto corre- function seudo–hazard − 1 18)adS and (1985) , distribution 1 ; 2 , , 0 . . . , 1 2 ≤ β , ido transformation of kind e , m. , . . . , facasclW classical a of 2 . . . , β ∈ ≤ R ; , 1 ALVIA nrdcdby introduced , EIBULL ssfcetto sufficient is 19)is (1996) EIBULL EIBULL (3.38d) (3.38b) (3.37d) (3.38c) (3.38a) (3.37e) model 125 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC iue3/5: Figure hc needn of independent — which oeprmtr fterflce W reflected the of parameters Some h orsodn aadrt is maxim rate type–III hazard the corresponding as The recognized been has distribution This hslast eeto ftecasclW classical the of reflection a to leads This 126 eutn in resulting f R ( y | ,b c b, a, aadrt fterflce W reflected the of rate Hazard h F R = ) R ( ( y y | | b c ,b c b, a, ,b c b, a,  c Var α α a sicesn,adge to goes and increasing, is — E 4 3 − = ) = ) b ( ( ( ( y Y Y Y Y 0 y y . ∗ 5 = ) = ) = ) = )       c b c −  = = EIBULL 1 exp a exp − b α a a − Var a  4 y − − − α  − ( EIBULL  ( 3 X − b b b X ( c itiuinare distribution  − X l 2) (ln (1 Γ ) EIBULL 1  = ) , 1 a 1 ) , a − , − 1 b − b − b 1 itiuin( distribution exp y 1 2 /c /c y  exp (Γ c ,  ) Faotavria xsat axis vertical a about DF   1 c 2 ∞ /c  −  − ; − for for  with Γ mdsrbto nSc.3.2.2. Sect. in distribution um a < y a  1 2 ) − a b , a y a, < y y − y b 0 = to ≥ ;  y c ,c> c b, a a. eae distributions Related 3   b , − c  .      1 = , 0; ) (3.39h) (3.39g) (3.39d) (3.39b) x (3.39e) (3.39c) (3.39a) (3.39f) (3.39i) = a Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h est ucini ymti bu etclln in line vertical a about symmetric is function density The n h D ssmercaottepit( point the about symmetric is CDF the and etae nterdcdfr with form reduced the on centrated ye hs B W Thus, double lyze. the version three–parameter general its In obeepnnildistribution exponential double n CDF and and 34a sdpce nteuprpr fFg / o eea v several for 3/6 Fig. of part upper the in depicted is (3.41a) itiuini ioa for bimodal is distribution 0 hc xetfor except — which obnn h lsia n h eetdW reflected the and classical the Combining 9 Chapters in discussed methods the by the ycagn h ino h aasmldfo eetdW W reflected classical a the from from sampled data data as viewed the of sign the changing By W the of Modifications 3.3 < c < obeW double f D ( y F h 1 D | D h oe ato i.36sostehzr rate hazard the shows 3/6 Fig. of part lower The . ,b c b, a, ( ( y y ALAKRISHNAN EIBULL | | ,b c b, a, 0 F = ) , D 1 c , ( y c = ) 2 = ) c | 1 = b distribution 0 ,

EIBULL          f 1          a sfrfo en oooe ti smerci n case. any in asymmetric is it monotone; being from far is — D c , > c − ( b 1 = ) y c /K − y 0 ( | − OCHERLAKOTA

. 1 0 exp 5 c distribution 0    , 2 y − or n nmdlfor unimodal and y c . 1 exp 5 ) 1 − a c , EIBULL c ihDF with L 0 − c exp 0 = 1 exp − = ) . exp 1 APLACE exp 5  − 1 ff. exp −   exp {− {− 0 and 2 c − −  . y exp[ 5 EIBULL  oe.Tu h aaeesmyb estimated be may parameters the Thus model.  y | ( a  {−

− − y = − c b a − | } y b c distribution y 1 = − y − ,F a, b 18) h nrdcdti oe,con- model, this introduced who (1985), − ) − − y b c 1 c − } y  } a y y EIBULL hn 34ab uninto turn (3.40a,b) Then, . exp c

y c oesit n itiuinrslsin results distribution one into models

c   D c  c c c 1 = ]

  −| {− 0 = ; ,a y, y for for for for for for n a nipoe oefor mode improper an has and . y itiuini o ayt ana- to easy not is distribution . 5 EIBULL = | .When ). c le of alues } ∈ a y y y y y y (see ≤ ≥ R ≥ ≤ ≥ ≤ ; 0 0 0 0 itiuin tcnbe can it distribution, a. a, ,c> c b, , . , , i.3/6 Fig. c c                      oieta the that Notice . 1 = ehv the have we , 0; for (3.40b) (3.41b) a (3.40a) (3.41a) (3.41c) 0 = 127 ) Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h bouemmnsof moments absolute The n h utssas kurtosis the and hs h aineflosas follows variance the Thus, aaee siainfrti itiuin—mil ae o based mainly B — by distribution sidered this for estimation Parameter h oet fti eue omo h obeW double the of form reduced this of moments The iue3/6: Figure 128 R AO /R AO /N 0 est ucinadhzr aeo h obeW double the of rate hazard and function Density ARASIMHAM ALAKRISHNAN b , 1) = E ( E Y Y α  (1991). 4 r | r aiyfudt be to found easily are ( Y /K = ) Y | OCHERLAKOTA r Γ = )     Var Γ = Γ  (  Y  + 1 + 1 + 1 Γ = ) 0 4 c r c r c     , + 1 18) R (1985), r , for for Γ EIBULL 2 0 = 2 c   r r + 1 , 1 even odd , AO 2 2 c itiuinaegvnby given are distribution . . . ,  re ttsis—i con- is — statistics order n , /N . . EIBULL    ARASIMHAM eae distributions Related 3 distribution 18)and (1989) (3.41d) (3.41g) (3.41e) (3.41f) ( a = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC a n h nes W inverse the and cal ol sddsrbtos e h pe atof (c part tail upper right the see long distributions, a used exhibits monly generally function density The engines. phenomen diesel degradation of describe crankshafts) to (pistons, model suitable a as (1982) otecasclW classical the to tion h neiro h upr,gvnby given support, the of interior the itiuinare distribution The hsdsrbto skonas known is distribution This n h CDF the and a h DF the has netn 34b ed otefloigpretl functi percentile following the to leads (3.42b) Inverting skewed. positively always is it and h bv nerli o nt for finite not is integral above The hsi osqec fteln,ftrgttail. right fat long, the of consequence a is This .. nes W Inverse 3.3.3 W the of Modifications 3.3 nSc.322 ehv hw ht—when — that shown have we 3.2.2. Sect. In W W 14 13 EIBULL EIBULL r (M h itiuini dnia otetp–Imxmmdistri maximum type–II the to identical is distribution The ugse edn o hsscin C section: this for reading Suggested 18) E (1989), t oetaotzr sgvnby given is zero about moment –th URTHY itiuin—tetasomdvariable transformed the — distribution distribution RTO f I opeetr W complementary ta. 04 .2)Tedsrbto a enitoue by introduced been has distribution 23).The p. 2004, al., et /R ( y E EIBULL APONE | ( EIBULL ,c b, Y EIBULL r = ) = ) (M 18) J (1984), EIBULL est h nes W inverse the density = UDHOLKAR c b y F itiuin o h aesto aaee aus.I cont In values). parameter of set same the for distributions P distribution  I b Z 0 nes W inverse ( ∞ r = y b IANG y distribution Γ  y | y F −  r r ALABRIA ,c b, EIBULL ∗ I c 1 − /M  ≥ − = 1 − exp = ) c b 1 Y ( URTHY c /K   P b exp ssc,when such, As . r c =  = ) EIBULL  OLLIA /P 13 + 1 y b X  distribution ULCINI /J for  c − i.3/7 Fig. b b  − I − 2 X 20) M (2001), c c  EIBULL − 94 and 1994) , − a −  c < r y b 1  1 ∼ distribution 18,19,19) D 1994), 1990, (1989,  ln /c bution. y b exp − sho ,  . We P c −  on UDHOLKAR
only  c est lashsamode a has always density 1 ; c  − ( (D /c igtedniiso h classi- the of densities the wing mae ihta ftecom- the of that with ompared ,b c b, a, ≤ . y  . RAPELLA ees W reverse . fmcaia components mechanical of a y b ≥ 2  h ainei o finite. not is variance the , . ) 14 − 0; i.e., , c /K  ,c> c b, te ae o this for names Other OLLIA d RAPELLA 1993), , y EIBULL X 0 (1994). a classical a has K ELLER 19) E (1993), reciprocal distribu- (3.42d) (3.42b) (3.42a) (3.42c) (3.42e) y tal. et ∗ rast RTO 129 in Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ihamxmmat maximum a with sa piedw ahu setelwrpr fFg /)adh and 3/7) G Fig. inverse of and part lognormal lower the the of (see bathtub down upside an is iue3/7: Figure uuesml rma nes W inverse an from sample future aees hyas tde B a studied also They rameters. h aadrt fteivreW inverse the of rate hazard The 130 C ALABRIA /P est ucinadhzr aeo h nes W inverse the of rate hazard and function Density ULCINI y h ∗ h 18,19)hv ple h Lmto oetmt h w p two the estimate to ML–method the applied have 1994) (1989, hc steslto of solution the is which , I ( y y lim → | 0 ,c b, 1 h AUSSIAN = ) I AYESIAN − ( EIBULL EIBULL y exp |  ,c b,  b c  y b   − lim = )  distributions: distribution itiuinudrtp– n yeI censoring. type–II and type–I under distribution 1 praho rdcigteodrdlftmsi a in lifetimes ordered the predicting of approach  c y b − y y b  →∞ −  exp c c −  1  h = exp I − ( y c   | 1 + y b c ,c b, −  −  0 = ) . c  y b EIBULL  sabhvo iia othat to similar behavior a as − c  eae distributions Related 3 distribution (3.42f) a- Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC itiuino h oainsaefml hc ses to easy parameters is the which estimate location–scale–family the of distribution FadH eogn o(.3)are (3.43a) to belonging HR and DF 31b)wt oainparameter location with (3.17b)) Provided, parameter scale and ehave we 34a sntigbut nothing is (3.43a) and variable ucinof function nodrt eietemmnso Log–W a of moments the derive to order In 34d a engahdin graphed been has (3.43d) Y transformat following the introduced have we 3.2.2 Sect. In .. Log-W 3.3.4 W the of Modifications 3.3 W 15 EIBULL scle a called is (1969). ugse edn o hsscin K section: this for reading Suggested a data. t fteoiia W original the of h ecniefnto is function percentile The . Log–W EIBULL F Pr X f ( ( L EIBULL t Y t ( | P t ,b c b, a, b F ≤ | EIBULL = L a and distribution t ∗ ( h 1/ Fig. = ) a t b , L ∗ | = ) EIBULL variable c ( ∗ a Z + = 1 = 1 = t = ) et pnapeeiglgtasomto fteobserved the of log–transformation preceding a upon rests ∗ Y | distribution b , = b a Pr ∗ ln( = ∗ ∗ Pr Pr c for b , b ( ) ln 1 X − − ∗ h D fatp––iiu itiuin(see distribution type–I–minimum a of CDF the , (   OTZ ∗ ait skontelgtasomto eut na in results log–transformation the known is variate Y  X trigfrom Starting . ln( = ) a exp exp exp b 15 − a ≤ X ∗ ∗ − ∗ /N − X 0 = ln(1 =1 := t =ln :=  − ln   1 = ) ADARAJAH EIBULL b a 1 − − − ∗ t a ≤ b − and = ) ); exp /c. exp − a b e ∗ e ) b a t − t P ≤ X   ∗ b   Y ) b ∗ c itiuin eitouetereduced the introduce we distribution, exp = − 
t t ≥ ( 1 = 0 ; 20) L (2000),  c − b − t Pr exp

b ∗  a. − ∗ a a  h aadrt sa increasing an is rate hazard The . − ∗ < P < X ∗ ln  elwt.S n prahto approach one So with. deal o of ion   t b ≤ − . ) IEBLEIN t b  ∗ a − a b + ∗ X . 1 a  .  e ∼ c t /Z   We ELEN , ( ,b c b, a, 15) W (1956), : ) (3.43b) (3.43d) (3.43c) (3.43a) (3.43e) (3.43f) HITE 131 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h orsodn a oetgnrtn ucinis function generating moment raw corresponding The sn 26c eesl find easily we (2.65c) Using with 132 E Z 4
= = Γ(1) = γ d 4 4 ( + Γ(1 E + d n γ Z θ M ψ 4 2 F 3 f π 4
L L Z E 1 6 + (1) θ 2 ( ( ( E ) z z θ + Z = Γ(1) = =

( = ) | | θ Z 2 0 0 =0 3
= ) , , 20 π = ( + Γ(1 = = − )=exp = 1) )=1 = 1) d Γ = ψ 4 3 γ = = Γ(1) = = = ( + Γ(1 2 − 3 −∞ Z E (1) 0 d Z ∞ −  (4) ∞  θ 4 ψ γ − ψ d d 3 u − ψ γ 2 ψ π γ (1) 3 e 1 Γ(1) (1) 2 ( + Γ(1 γ θ 1 3 + (1) e z θ ( + Γ(1 + 2 ′ θ e 1 3 + (1) Z θ − ≈ − d d θ ) ′′ − 2  π θ θ )

(1) exp ψ u θ  + . exp 2 2 z =0 2 d / θ 0 1 + (1) u, ψ − 6 θ ≈ . ) ψ 577216; ) ′′ Γ = z

≈  e θ (1)

(1) setting − 23 ψ − =0 z θ 1 =0
′ e . (1) ′′′ . ψ . e 5615 ψ 97811; − ≈ z Γ = − (1) ′
Γ = (1) ′ z z ,  1 + (1)
2 u ′ . 5  4 + d (1) ′′ . z, = 44487; ∈ (1) ψ R e ψ z ′′ , (1) , (1) eae distributions Related 3 ψ  ′′ 1 + (1) ψ ′′′ (1) (3.44d) (3.44g) (3.44b) (3.44e) (3.44c) (3.44a) (3.44f) o Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ncmo agaetetowords two the language common In itclsinete aedfeetmaig.W onC join We meanings. different have they science tistical W Truncated 3.3.5 mig one hand other misapplied. the is on “log-normal” but chosen, been have could variable ees fta ewe omladlgnra aibe.Th variables. log–normal and normal between that of reverse X that efial oeta h eainhpbtenW between relationship the that note finally We hc mle that implies which aaee siaino h Log–W the Sect. of in estimation detail Parameter greater in statistics order to revert will We h siainfreteevledsrbtos D distributions. value extreme for estimation the u tl oeotycs td pligteLog–W the applying study case noteworthy still but itiuinfrbl ern a oeb L by done was bearing ball for distribution K by monograph moment the of in provided is method inference the statistical by estimation investigated have (1994) oteLog–W the So 34cd edt h olwn eut o eea Log–W general a for results following the to lead (3.44c–d) W the of Modifications 3.3 otnigwt h eue Log–W reduced the with Continuing ftefis re ttsi sml iiu)i apeo si of sample a in minimum) (sample statistic order first the of 16 scle o-omlvral,wieif while variable, log-normal a called is Y ugs edn o hsscin A section: this for reading Suggest 19) S (1993), M saLog–W a is ARTINEZ HALABY /Q EIBULL UINTANA EIBULL Z /A Var 1: n E L est sngtvl kwdwt mode a with skewed negatively is density –Y Pr EIBULL ( ( a h aedsrbto as distribution same the has Y Y α α EIBULL 19) M (1991), OUSSEF aibe ehp oeaporaenm hnLog–W than name appropriate more a Perhaps variable. 3 4 ( = ) = ) Z Var 1: − ≈ ≈ n E Z ≤ distribution Z Z 19) S (1992), 1: distributions ROIAN 5 a C 1: 1: n t b . E ∗
EIBULL n n 4 1 ∗ min = EIBULL WEN + truncation ) . .

13955; 2 1 = 1 = = 1 b π 16) C (1965), ≤ ∗ = = /P UGIURA 2 i exp( ( ≤ / IEBLEIN ARRESOL − F 6 n itiuincnb iwda pca aeof case special a as viewed be can distribution L − − − Var ( γ aibe ecnsaetefloigproperty following the state can we variable, ≈ Z  Y ) EIBULL γ t  exp[ i /G 16 ≈ 1 ROWDER and ( ); ) 1 ln( + − EKKERS Z . a W a has 494( 644934 OMI − a /Z = ) ln( 19) M (1991), Z − Z ∗ censoring F i ELEN − e n 18) W (1985), OTZ EIBULL n L n − iid n Log–W and π ) ( ) 0 19) G (1990), 5.  t ln( t OHEN 2 . ; , ] EIBULL | / ta.(99 n C and (1989) al. et 577216 /N b 0 EIBULL 6 (1956). ∗ n , ITTAL . ti,i log if is, at ze ADARAJAH ) ) 1) 2 n hence and , INGO itiuina h lifetime the as distribution .Mr nomto about information More s. tas ru httename the that argue also ht ; r yoys u nsta- in but synonyms, are n  19,p )we esay we when 1) p. (1991, ROSS y n /D b ∗ itiuinte esay we then distribution EIBULL ∗ 18,18,1998). 1989, (1988, variable: AHIYA = ; 17) H (1971), a ∗ X 20) h first, The (2000). n leptokurtic. and 18) S (1989), aibe sthe is variables snra then normal is HRISTOPEIT WANG EIBULL HALABY (3.46b) (3.45b) (3.45d) (3.46a) (3.46c) (3.45c) (3.45a) (1996), 133 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC hsi lorfre oa the as to referred also is This olwn pca ae f(3.47a): of cases special following rnae itiuini ohn u odtoa distr conditional a but nothing restrictin is of distribution thus size truncated removed the A been to has population according original — the where of one is population truncated A truncation 134 curn hntevraei ieieo uain are duration, a or mos lifetime The a only. is range variate the restricted when a occurring in observable is variate the oe sgvnb h olwn CDF following the by given is model trigwt he–aaee W three–parameter a with Starting censored. been has sample inferent the the how because and testing and type estimation used to commonly turning the Cen fore present (3.48a). will We see 1; life–testing. 1 to in amounting than used DF less integrated be an would at which elimi arrive population, to is the units of of size portion reduced certain the a v population exact truncated the a by In not but number, their omi by not presented are an are units and identified sample known be censored may the So space measured. sample otherwise the of areas restricted some esrdsamples Censored 17 F • • • 2.6 e Sct. See DT left–truncation oelf rnainpoint truncation left some bv oergttucto point truncation right some above noiia itiuinaesbitdt oekn fscree right–truncation of kind some to lasting burn–in submitted are distribution original an ainaeshdldt ei s o aiu ieof time maximum a for use in be to scheduled are lation double–truncation oto falitems. all of portion ( x | sanotion a is ,b ,t c, b, a, . ℓ r hs nwihsml pcmn ihmaueet htl that measurements with specimens sample which in those are t , t ℓ nsc oe rnae itiuintesalrrealizat smaller the distribution truncated lower a such In r eae opopulations to related = ) nt ftime. of units nuprtuctddsrbto smsigtegetrreal greater the missing is distribution truncated upper An obytuctddsrbto nycmrsste“middle the comprises only distribution truncated doubly A = = obytuctdW truncated doubly 1 exp F 1 exp F t − ( − ℓ ( t x aebe mte.Ti ilhpe hnaliesof items all when happen will This omitted. been have  r  EIBULL exp exp | | − − ,b c b, a, ,b c b, a,     t r t t ℓ hsmgthpe hnteieso popu- a of items the when happen might This . ℓ ) ) t t − b − itiuin h eea rnae W truncated general the distribution, b ℓ ℓ − − − b − b a a F whereas F   a a c ( c (   t   t ℓ c ℓ c | EIBULL − − | − − ,b c b, a, ,b c b, a, exp  exp  ibution, censoring a prahdpnso whether on depends approach ial t x h upr oasalrrange. smaller a to support the g r ae ihu elcmn thus replacement without nated   ) ) − − b b aiirtpso truncation, of types familiar t fcnoigi et . be- 8.3 Sect. in censoring of s oe ape r extensively are samples sored ; − − distribution tdo ogte u hyare they but forgotten or tted a a t r10,hst erescaled be to has 100%, or le ftercharacteristic. their of alues hscutd u r not are but counted, thus d a  r   17  h aibe—sm part some — variable the c only. c ≤ t x   ig .. elzdb a by realized e.g., ning, r h odto en that being condition the − is t − b b . ℓ eae distributions Related 3 eae osamples to related a ≤ a   c x c   entc the notice We . ≤ t r osbelow ions < izations EIBULL ∞ (3.47a) , ein ie ” . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC terval h Fblnigt 34a is (3.47a) way. to trivial belonging a DF in The result truncated doubly the distributi from truncated follow right tr not and doubly left the the to for pertaining results results Special give will we sequel the In h hp of shape The h aadrt eogn o(.7)is (3.47a) to belonging rate hazard The where . oictoso h W the of Modifications 3.3 • • • f • • • DT h ( ( ( t > c > c ( x DT ℓ t t < c t ℓ ℓ ℓ ( ∗ ≤ x = = ,t a, > = ( 1 1 x | x 1) ,b ,t c, b, a, ,a c (  (see x x x < x < ( ℓ ∞ r c t , ∞ | | ℓ ,b ,t c, b, a, < ,b ,t c, b, a, − t , 1 ) r F ) ∗ i.3/8 Fig. ∗ = ) ed oteoiia,nntuctdW non–truncated original, the to leads RT and r 1) ∞ ie the gives t < EIBULL = ) = /c = = ) ( ⇒ x r t ie the gives = =  ℓ ℓ ℓ | 1 = , t , ihtefloigresults: following the with ) /c x > ,b ,a t a, c, b, a, f exp F c b ∞ ⇒ DT r distribution 1 F 1 ( stemd ftenntuctddistribution. non–truncated the of mode the is )  1 = ) ettuctdW truncated left t − − ∗ (  r sdtrie ytesaeof shape the by determined is ( f f b c x t = ) · | DT r 1 − ( ) F F ,b c b, a,  − x ih rnae W truncated right b | f − sincreasing, is DT  ( ,b c b, a, DT f | ( x ⇒ − x a · ,b c b, a, r exp ( f t ) −  | ( = ) x b ℓ exp ( ( c x ,b c b, a, sunimodal, is ) f x − x − b | a − | DT  ) ,b c b, a, 1 | |  ,b ,t c, b, a, a  ) ,b c b, a, − 1 ,b c b, a, c 1 exp F  − ( − ) t − c · F ( 1 ) ℓ  t t ) F exp (  ℓ ℓ − exp sdecreasing, is exp b EIBULL x ) ) − | − ( b ,b c b, a, ℓ t a | n ilb ie nywe hydo they when only given be will ons t , r exp ,b c b, a, t   a   ℓ | c r 1  − − − ,b c b, a, − EIBULL b ) c − −  nae W uncated ; )    − a ;  − distribution ) F a x  t x  EIBULL r t c ) a  ( ≤ − − r b x b − x b − − ≤ t − b f | t r a a − distribution b a  ℓ ,b c b, a, F ( t −  b  a x  ≤ ℓ a c x ( c  c EIBULL   x | ≤ a   c distribution. − ,b c b, a, x b c   |  ) x ,b c b, a, c . ≤ a  ihCDF with . ≤  t c ) r  t distribution. r vrtein- the over ) < ihCDF with . < . ∞ ∞ (3.47b) (3.48b) (3.47c) (3.48a) ; ; 135 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC oet a eeautdol ueial.Loiga h re the at Looking numerically. only evaluated be can moments where iue3/8: Figure h rtfco ntergthn ieo 34b stehazar the is (3.48b) monotonical a of ing is factor side second right–hand The distribution. the truncated on factor first The 136 ahu hp ilcm pi h obyadrgttuctdc truncated right and doubly the in up come will shape bathtub l rnae W truncated bly h ecniefnto eogn o(.7)is (3.47a) to belonging function percentile The M E C F ∞ E DT X − WEN 1 γ DT as r ( [ P · x
| /P = ) · → ] este ftuctdadnntuctdW non-truncated and truncated of Densities ARRESOL = steicmlt am ucin(e h xussin excursus the (see function gamma incomplete the is a t r EIBULL + So . ( 1 b γ − ( " exp h exp ln r 19)gv h olwn oml o h a oet fted the of moments raw the for formula following the give (1991) DT " − distribution:  c  1 ( j · − ) t 1 + − ℓ a eete eraig nraigo ahu–hpd Th bathtub–shaped. or increasing decreasing, either be can F − b t ( r

t a ℓ −  b  | − c ,b c b, a, a   t r c  − b ) X j a − =0 r  P c  #  j r − F  ( γ 1 b t " r r − r | EIBULL j ,b c b, a, − yicesn ucin approach- function, increasing ly c a j j × 1 + ) sswhen ases ue W duced ae(.6)o h non– the of (2.36b) rate d − distributions

F  ( t eae distributions Related 3 t 2 Sect. ℓ ℓ | − b EIBULL ,b c b, a, 0 a < c <  .S,the So, .91). c #) )  variate #) , 1 (3.48d) (3.48c) 1 . /c . ou- U e , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC and and ie nitrsigapiaino h rnae W truncated the of application interesting an gives swl steukonlf on ftucto ymxmmlik maximum by truncation of point left unknown the as well as yM by obndi n a raohrweeya es n fteeva these of one least at W whereby be another to or way vari one several in when combined result which models presents distributions section more This or two including Models 3.3.6 systems. missile of aits hn(et ...)w unt h ieiedistr lifetime the to turn we 3.3.6.2) (Sect. Then variates. iodapoc n yS by and approach lihood ntelf rnae ae(.8,)tr into turn (3.48d,e) case truncated left the In hntedsrbto srgttuctdw have we truncated right is distribution the When etn yohsso h aaeeso rnae W truncated a of parameters the on hypotheses Testing i.e., W the of Modifications 3.3 C B W S UGIURA AYES ROWDER 18 EIBULL E a A ITTAL 0 = ROIAN X C See approach. EIBULL RT r /G E and itiuini oprsnwt h obytuctdnorma truncated doubly the with comparison in distribution /D 19)adM and (1990)
16)gvsa vlainbsdo the on based evaluation an gives (1965) OMI E U = AHIYA DT r b X ewl rt(et ...)suytedsrbto ftes the of distribution the study 3.3.6.1) (Sect. first will We . 1 1 = LT r
− 18)gv oetrtodarmsoigtergttrunca right the showing diagram moment–ratio a give (1985) E =
34d eoe itebtsimpler: bit little a becomes (3.48d) , exp U 18)adW and (1989) 1 LT r exp = E − HARENKVIC  EIBULL ARTINEZ exp
HALABY U − exp exp =  RT 1 r  { Γ {− t t
 ℓ c  r } distribution = − r b t { r c /Q − INGO } c t t 1 a 19)adS and (1993) ℓ ℓ c UINTANA   } − j − b c  γ  1 + exp a  Γ 18,18,19)apyn h aiu like- maximum the applying 1998) 1989, (1988,  X j r c 1  18 =0 c  r  /C {− r 1 + c χ −  1 + X j 2 OHEN t =0 19) aaee siaini discussed is estimation Parameter (1991). –distribution. r j r γ r c

 }  t HALABY   r c EIBULL r γ b  − j r r  − c − 18)fretmtn h parameters the estimating for (1984)  − r c γ j j b γ  a 1 + EIBULL r 1 +  j r c − γ /A r c tso eea itiuin are distributions several or ates j 1 + oe nrlaiiyprediction reliability in model  bto fasse aeup made system a of ibution

a 1 +

r L t j r c  –Y − ×  c

itso itiuin has distributions or riates t lho.H elihood. itiuini rae by treated is distribution t

. ℓ j ℓ c OUSSEF t  − ℓ c b 1 +  a distribution. l .  .

c mo W of um   19)uiga using (1992) t WANG r − b a  EIBULL (3.50b) (3.49b) (3.50a) (3.49a) (3.48e) c (1996)  137 ted Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h umto n(.1)rn vrall over runs (3.51c) in summation The r Assuming of variance The hi itiuin.W ilol eoto h u fidW iid of sum the on report only will We distributions. their the of distribution The W 3.3.6.1 mixin continuous practice. Section as and theory well in as components. intensively discrete explored failing with randomly and more composing with or two linking by 138 so osdrbeiprac nrnwlter Sc.4.4) (Sect. theory renewal in importance considerable of is of sum a on results general some summarize will we First, 4.3). (Sect. with R nteitgadaedfie.Terwmmnsof moments raw The defined. are integrand the in i h rao nerto n(.1,) otisalargumen all contains (3.51a,b), in integration of area the , ≤ n i aits h FadCFof CDF and DF The variates. iid r h ubro umnsis summands of number The . EIBULL E Y ( Y F is f r n n = ) folding ( ( y y = ) = ) = u of sum X X E f 1  i ( ( R R ∼ Z Z · X w rmore or two = ) r Var 1 1 We f f ! E + ( ( r x x f ( ( 2 ( Y Y X ! ) ) Y r ( r ,b c b, a, + · r . . . Y ! F f = ) = ) ) 2 n n n r ie ytefloigrecursions: following the by given are r n + − –tuples − = − and 1 1 n 1 ); . . . X X
i i ( X i ( ! =1 =1 n n y npriua ehave we particular In . =1 n y variates E − i + − E Var F X 1 = X ( ( x 1 X Y x i X r ( 1 r ) ) ( · 1 1 n d X = ) , i r ie by given are r , d
) ) x 2 x r . E i n, , . . . , sfudb ovltn rfolding or convoluting by found is 2 ; )  ; r , . . . , . F X n n ( sfrwihteDsadCDFs and DFs the which for ts 2 r 2 = · 2 = 2 ) EIBULL n o W for and
. n . . . , ;tpc hc aebeen have which topics g; , ) 3 3 ... o3365deal 3.3.6.5 to 3.3.6.3 s with . . . , . . . , E X eae distributions Related 3 aits uhasum a Such variates. ; ; i n P r =1 n EIBULL n
r . i = r processes and (3.51d) (3.51b) (3.51c) (3.51e) (3.51a) 0 ≤ Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 F © 2009 byTaylor& FrancisGroup, LLC unn oagnieW genuine a to Turning n For necpini h case the is exception An oepeieya E an precisely more The variates. distributed exponentially dently W for (3.51d,e) of evaluation the Whereas onteitcoe omepesosfrteintegrals the for expressions form closed exist not do o vni h case the in even not have 35ab uninto turn (3.51a,b) W the of Modifications 3.3 f 2 = c F f n n 19 2 = 2 ( ( 2 ( y y ( n y y = ) = ) summation. nSc.314w aearaymnindta h W the that mentioned already have we 3.1.4 Sect. In | ∈ | and 0 0 b, , N b, , c , a a n a n Z Z F )= 2) f )=1 = 2) y y 0 = 2 2 1 = ( ( c b b c y y   | | when ,b, a, ,b, a, and x x RLANG n − b − exp b 2 = a − )=1 = 1) )= 1) F f a c a EIBULL c n n 0 =    exp 2 = EIBULL ( ( 1 = c . − 2 y y c − − b itiuinas distribution | | ehv h adoeformulas handsome the have we , 1   1 3 exp 0 0 exp or − o all for y b b, , b, , y  itiuin i.e., distribution, c  distribution  2 −  − )=1 = 1) )= 1) 3 =  b y b − − 2 exp  2  n 2   a . 2  ∈ x  y b x − exp − N − b − b EIBULL exp b n y ..tecs ffligietclyadindepen- and identically folding of case the i.e. , − a n  a − − sapstv nee.For integer. positive a is −  √ (  y n b c n  exp c n 2   fl ovlto sagmadistribution, gamma a is convolution –fold 2  − − − c f a π F y 1 EIBULL 2 1 n b    1 6= n 1)! aitsi ut ipeteegenerally there simple quite is variates − − exp 19  − − b 1 1 y b ( 2 y ntergthn ie f(3.52a,b), of sides right–hand the on y b ewl nyeaut 35ab for (3.52a,b) evaluate only will we , ( y exp   a y  − 2 − 2 1 itiuini o erdciethrough reproductive not is distribution  −  n X i =0 2   − x b r x a − 1 y ) b y , ) d b  + π 2 y b d y i x 2 x  i i  ; b erf ≥ ! ; ,  ( . n b  2 n 2 y , 2 = a, b n − 2 = √ y 2 = ≥ y , 2 2 , 3 2 )  3 . . . , . . . . , a. erf and  ; c b 1 = (3.52b) √ (3.52a) y 2  139 we Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC erf iue3/9: Figure hc srltdto related is which 140 c=3 F f 2 2 ( ( · ( y ) y sthe is | | 0 0 b, , b, , )= 3) )=1 = 3) G este fatofl ovlto frdcdW reduced of convolution two-fold a of Densities AUSSIAN Φ(  exp z 4 16 − ) b h D ftesadrie omlvraeby variate normal standardized the of CDF the , ro function error Φ( 6  b exp − 9 − / τ 2  = ) ( 4  y y b − 5 y b  2 / erf  2 3 1 2  ) y b 3 h ( (  τ + 1 3 + 3 ( 6 = )  : y b − erf y √ 6 ) exp ] 2 ( 3 π τ/ erf / 2 Z ( 4 0  √ ( " τ − y y b 2) √ e 3 2 4 1 − − 3 i ) t  2 3  2 / y for d b 2 y b b t,  3  3 + ) 3  τ / 2 r ≥ #) EIBULL √ π 3 0 3 . (2 π b exp 3 eae distributions Related 3 variates 3 +  4 3 y  3 y ) b  erf 3  " × (3.53b) (3.53a) √ 2 3  y b  3 / 2 # Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC aallsse swriga oga tlatoeo its of one least at as long as working is system parallel rate. A hazard component’s each of sum the i.e., al hnaliscmoet aefie.Tefiuedistr failure The failed. have components its all when fails hs h Ro eissse ipyis simply system series a of HR the Thus, F ahcnitn of consisting each w ipessesi eiblt hoyaethe are theory reliability in systems simple Two W 3.3.6.2 W the of Modifications 3.3 Pr eissse swriga oga l its all as time long of probability as survival working is system series A of 35a od nyfridpnetyfiigcomponents. failing independently for only holds (3.54a) obndit t h eissse’ alr distribution failure system’s series The it. into combined ihDF with i ( ( n X t = ) with i t > Pr R ) ( stepoaiiyo t uvvn ftime of surviving its of probability the is S EIBULL X ( i t ) ≤ ≤ n t R ) opnnsta ilfi trno.Frcomponent For random. at fail will that components oesfrprle n eissystems series and parallel for models R F etepoaiiyo alr pt time to up failure of probability the be i f ( F P S t S S ) ( ( ( EIBULL ( t t t ∀ t = ) = ) = ) 1 = ) t i h o eissse smr eibetels opnnsare components less the reliable more is system series a So, . sgvnby given is = = S = ( t distribution = ) − Y Pr i Y Pr i X R i =1 =1 n n =1 n Y i S ( ( = =1 n X X ( F R    t i 1 1 ) i R ( ( j X X i t R i ≤ ,X t, > f t =1 i =1 =1 ) n n ) ( Y S S . t n ,X t, ,j ( 1 = ) ( h t R n t 6= f ) ) i i i ( i opnnsaewrig ota h series the that so working, are components ( (  t 2 t 2 = t ) 1 ) ) , ≤ eissystem series ,...,X , . . . t, > − − . X i Y ,...,X , . . . t, i =1 t n F =1 n . j ( R  f t olw as follows 1 ) i i bto D sthus is CDF ibution (  ( − R t    t n ) ) S t F then , opnnsi okn and working is components f n n ( i i t ( ( ) ≤ t > t n the and t ) ) sadcesn function decreasing a is  , t ) ) R i ( t 1 = ) aallsystem parallel i (1 ≤ − i ≤ F (3.54d) (3.54b) (3.55a) (3.54c) (3.54a) i ( n t ) = ) 141 let , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC .. Ri egtdsmo h niiulhzr ae,whe rates, hazard individual the of sum weighted a is HR i.e., h Rflosas follows HR The eraigwith decreasing with eoeeautn 35ad n 35ad sn W using (3.55a-d) and (3.54a–d) evaluating Before inta h w rcdn ytm r odriecsso t of cases borderline are systems preceding two the that tion ikmodel risk system nsre ytmi “ a is system series an h eiblt ucino “ a of function reliability The efis vlaetesre ytm lokonas known also system, series the evaluate first We htexactly that m Z o neednl aln units. failing independently for 142 ucino uvvlfnto is function components survival more or the function reliable more is system parallel a Thus, n t Fis DF its and ( 20 o–alduisadteohrclass other the and units non–failed ,n m, non–W oesfrwihoecmoethsaW a has component one which for models M t R . URTHY ,wihi okn sln sa least at as long as working is which ”, k,n ) stesto l attosof partitions all of set the is ( EIBULL t ihW with , m = ) /X fthe of IE m t X /J hssmi utpidb h factor the by multiplied is sum This . . n = IANG k EIBULL   n n R h Z –out–of– nt aentfie.Teotrsmrm vrall over rums sum outer The failed. not have units 20)suymlirs oeswt nes W inverse with models multi–risk study (2004) P P X ( m,n ( ( f t t P 1 = ) = ) ) ( k itiue ieie fiscomponents. its of lifetimes distributed t R = ) –out–of– = = F i n 1 P ( ytmadaprle ytmi “ a is system parallel a and system ” − = t ( ) R R R F F t f Y i ) · =1 P P P n n P P P n X F sadcesn ucinof function decreasing a is i ( . . . ( ( ( ( ( =1 nt notocasssc htoecascomprises class one that such classes two into units n − EIBULL t P n t t F t t t ) ) ) ) ) ) ytmi ie by given is system ” ( m i    ( t X X i i · ) t =1 =1 = n n R alduis ninrsmgvsteprobability the gives sum inner An units. failed 1 = ) j X i =1 =1 n i k Y itiue ieieadtermiigcmoet are components remaining the and lifetime distributed h m R 1 f n ,j i fits of ( i i − ( F f F ( 6= t ( t t ) − i t i ) P i ) ( F ( ) · F EIBULL t t R ( F Y i P ) F ) R t j F =1 n n i i ) ( ( i ( . ( i optn ikmodel risk competing i m t t t ( opnnshv o ald Thus, failed. not have components t ( ) ) F  ) +1 t ) t    1 ) ) P X i , =1 emr eea “ general more he ( n − ( f t eteweights the re itiuin,w hudmen- should we distributions, t ths h ytmsreliability system’s The has. it ) i ) R ( F  f · t EIBULL i ) i i R . . . ( ( ( n t t t P ) ) ) with  ( t · ) eae distributions Related 3 20 1 F itiuin n hybrid and distributions –out–of– hc sincreasing is which i F n Assuming ( P m t R ( ) t   k i ) ≥ ( –out–of– . t ≤ ) k n  or . system. ” F F i (3.55d) (3.55b) i (3.55c) ( multi– X ( (3.56) t t ) i ) ∀ n are ∼ iid i – . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC 1 u of mum 35a sesl eonzda h CFo variate a of CCDF the as recognized easily is (3.57a) 1 o,tesaeprmtrof parameter scale the Now, xoetae W exponentiated ntegnrlcs,i.e., case, general the comp In the of parameters location the and/or parameters shape 35a h parameter the (3.58a) c ta.(95,M (1995), al. et alda W called with system parallel A a 35a saseilcs fthe of case special a is (3.58a) aaee siaini hsgnrlcs setnieydi extensively is case general this in estimation Parameter 20,p.1619 n yD by and 186–189) pp. (2004, We W the of Modifications 3.3 W M n , , poly-W UDHOLKAR EIBULL 2 2 ecnsaesm eut ntehzr rate hazard the on results some state can we ( • • n , . . . , n , . . . , ,b c b, a, h aadrt o h opnn ihtelretsaeparamet shape largest the with component the for rate hazard h smttcpoete are properties asymptotic The o small For opnn ihtesals hp aaee.Frlarge For parameter. shape smallest the with component S utpiaieW multiplicative ⋄ ⋄ ⋄ ⋄ ⋄ n ( EIBULL t ) with ) ; i W iid h h h h h ; 35a un into turns (3.55a) , a aeol n ftrepsil shapes: possible three of one only have can i S S S S S .. h cl atri ayn with varying is factor scale the i.e., /S ( ( ( ( ( 1 = t t t t t ) ) ) ) ) UDHOLKAR R RIVASTAVA t EIBULL EIBULL sdcesn when decreasing is sicesn when increasing is sbttbsae when bathtub–shaped is ≈ ≈ , S model h aadrt ftesre ytmi erytesm sthat as same the nearly is system series the of rate hazard the ( 2 h h t n , . . . , exp = ) 1 n n ( ( t t X ) . sapstv nee,i ssbtttdby substituted is it integer, positive a is ) EIBULL distribution. aits(see variates F EIBULL i EIBULL for R for P T ∼ /H 19)adaaye nsbeun aesb M by papers subsequent in analyzed and (1993) S ( ; AVISON  t is ( We 35a un into turns (3.54a) = ) t UTSON t t − exp = )  distribution → ∞ → xoetae W exponentiated n ( i P model =1 a  n  itiue ieie fiscmoet ssometimes is components its of lifetimes distributed 0 i 1 1 b , , c /L t b − 1 c < − et 3.1.4 Sect. i − . i 19)adJ and (1996) b OUZADA c , c < ( c exp  e 35a.Assuming (3.55a). see , a − i c 1 −  ) 1 2 c ( c < 1  n needn,assuming independent, and  c < . . . < t /c c < . . . < − − h . exp = 2 S i T  –N a .Assuming ). ( c < . . . < n h eissse’ lifetime system’s series the and ) t t a oW no has c ) EIBULL ETO − IANG X b i f(3.54d). of =1 n  a j T n − −  cse yM by scussed 20) h alti distribution this call who (2000), b 1 1 < c /M i c  ∼  < ) n nns ieie r differing. are lifetimes onents’ 1 n b EIBULL . URTHY distribution n . t We . 1 − . − X c < t 1 ( ti prxmtl the approximately is it , a i /c X ,bn b a, ν er. j i  ∼ iid itiuinwe the when distribution ∈ c < . . . < c 19) hra in Whereas (1999). ∼ iid  URTHY − R We . c 1 We 1 nrdcdby introduced , /c oarv tthe at arrive to ( c < c , ,b a, ( /X UDHOLKAR ,b c b, a, ) h mini- the , n 2 i IE . c , < . . . < T /J (3.57b) (3.57a) (3.58a) ); o the for ); sstill is IANG i i 143 = = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC f ti oeotyta h feto h product the of effect the that noteworthy is It tdW ated h Rblnigt 35a is (3.58a) to belonging HR The o small For .. h ytmshzr aecnb prxmtdb h haz the by approximated be can rate hazard system’s the i.e., .. h prxmtn aadrt sta fteunderlyin the of that is rate hazard approximating the i.e., uinwt hp parameter shape with bution hp fteH sa olw as e h oe ato i.3/ Fig. of part lower the see (also follows as is HR the of shape h Fblnigt 35a is (3.58a) to belonging DF The 144 h P f P ( P ( • • • • • • t ( t ) t = ) oooial eraigfor decreasing monotonically for nmdlfor unimodal oeapoiae by approximated mode oooial eraigfor decreasing monotonically oooial nraigfor increasing monotonically nmdlfor unimodal ahu–hpdfor bathtub–shaped Fg 3/10 Fig. of part upper the (see is = ) EIBULL = = c n t ehave we n 1 t 1 = f n n ∗ − b c f itiuini h aea htof that as same the is distribution b c = P X  , F  ( a ( P t x t < c x ) > c n + ) ( − t b −  b ) F b a 1 a ( X   > c and ( c 1 2 c t − 1 − ) c n 1 "  h 1 ih—acrigt M to according — with > c n n 1 p P exp h − exp and n cl parameter scale and ( P 1 c t ( ) (  t  c c c n c ≈ ) < c ν 1 − − ≥ − ≈ ≤ ,   ) ≤  8 1  1 c n 1 x b n x and 1 c b and − 1 −  b 2 +  − ν , with b n c  n c   a c n 1 a c n  t ∈ t  n c − c ntesaeo h Ffrteexponenti- the for DF the of shape the on f c − c −   b ≥ b   R ≤ P exp ntesml W simple the in 9 + a a 0 = (0) + 1 1 b 1   . 1 , o large For . ,  c c n − UDHOLKAR − − n c − 1 − exp exp , ∞ 1  2 W g ) , r aeo W a of rate ard x  for −  − 10): b − − 1 EIBULL < c n t a  −  ehave we  EIBULL /H x c x n 1  eae distributions Related 3 − UTSON b − #) b 1 n a a itiuin The distribution. and n   c . c distribution.  EIBULL  f 19)—a — (1996) P n n − 0 1 = (0) − 1 1 (3.58b) (3.58d) (3.58c) . . distri- /b Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC where aaee ausaentes ofid hntecmoet all components the j When < i find. to easy not are values parameter h Fcnb eraigo a have may or decreasing be can DF The W multiplicative a on results General with h ecniefnto ssimply is function percentile The h a oet eogn o(.8) u with but (3.58a), to belonging moments raw The iue3/10: Figure W the of Modifications 3.3 W EIBULL F if 0 k ( c stenme fcmoet ihdsrbto dnia to identical distribution with components of number the is t i ) = distribution stetoprmtrW two–parameter the as este n aadrtso utpiaieW multiplicative of rates hazard and Densities c j M , µ F r ′ URTHY P − = 1 ( b n ( P a F EIBULL i = ) r /X P 0 = Γ ( c IE  t 0 a ) + 1 /J = F + ≈ ∀ distribution IANG P X i b (1 ( i =1 r c n ) t EIBULL n )  − n assuming and − c ≈ n X j i 20,p 9/)saetefloigresults: following the state 198/9) p. (2004, EIBULL =0 − k k ln F and + ) 1 oe.TeH a ee aeabttbshape, bathtub a have never can HR The modes. (  0 − 1 ( D having CDF t 1) − F k ) b 0 j P o small for oe ihcmoet aigdiffering having components with model 1  = a 1 ( /n n t 0 = X ) i c =1 − n  j 1 o large for o are , 1 ≤ 1 b /c  i c i t /c c , ( 2 EIBULL j 0 0 , ≤ 1) + < P < t, . . . 1 1+ F aeatwo–parameter a have models ≤ 1 r/c ( 1 t c . ) n . having and ( a b 0 = i c ≥ 1 b , (3.59b) (3.59a) (3.58e) and (3.58f) b j 1) = 145 for b 1 . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 f © 2009 byTaylor& FrancisGroup, LLC A h quantities The are hto osrcigasln ucin nrlaiiytheo reliability In function. spline a constructing criteria. of that smoothness distribu some more suffices or distribution two resulting of pieces together knotting by structed feetoi ue,i rmrl sdt riea non–mon a at arrive W to one only used in primarily encountered not is tubes, electronic of ... opst W Composite 3.3.6.3 aaees The parameters. ietbe stebhvo ftecniinlprobability conditional the of behavior the as life–tables oiedsrbto having distribution posite onW join where An ihcrepnigHR corresponding with nteinterval the In eraigfloe by followed decreasing u a orpsil hps(eraig increasing, (decreasing, shapes possible four has but 146 W E nc LANDT h EIBULL ( opst distribution composite nc x n τ 0 = ) fl opst W composite –fold ( x F = EIBULL = P = ) i –J ( i f a x F c b = itiuin rtitoue yK by introduced first distribution, i OHNS i i 1 nc ( |  1 x a and F ( i − x [ | x f i τ τ b , n te itiuin noas–aldhbi opst m composite hybrid so–called a into distributions other and a ( i = ) i i − n b τ F − ( i i τ r h nt rpit fcmoetpriin eas h e the Because partition. component of points or knots the are i i b , x h c , fl opst W composite –fold 1 i n a | ( P ; | i a /J x i i F τ c , a ( 1 = ) =  i i t | OHNS i i b , c ] ) ( i b , i a k h portion the ) EIBULL x − ∞ EIBULL i ≈ i i c , b , 1 | oa) h smttso h Rare HR the of asymptotes The modal). exp c , lokonas known also , o a for , a          n i i − i i ) c , ) b ,  − ≥ exp h h i − EIBULL i ) c , 18,Catr75.W ilpeetteW the present will We 7.5). Chapter (1980, 1 0 F distributions D sdfie as defined is CDF 2  ( ( = i n t i t  − ) = ) W = ) x fl opst oe hr r only are there model composite –fold h 1 − − ( for i EIBULL b EIBULL τ ( i  i x itiuin h ahu hp,rglryfudin found regularly shape, bathtub The distribution. c b b a c − 1 1 0 0 x i | τ 1  a i − |   b − c i etoa model sectional a i i b , 1  b c a i AO t t − Fis DF 1 0 uppltos fcus,i spsil to possible is it course, Of subpopulations. ≤ i i c , for   1  b , c c x c 15)t oe h ieiedistribution lifetime the model to (1959) i 1 0 i ) i − −  ≤ − τ 1 1 for i 1 − sthe is τ c , k 1 i for for ; fdaha age at death of i ≤ τ − oa olwdb nraigor increasing by followed modal yadlf etn h composite the testing life and ry i i h aki oehtsmlrto similar somewhat is task The − 1 1 = in nsc anrta the that manner a such in tions x ); i 1 t opnn nCFform. CDF in component –th t t ≤ or ≤ i → ∞ → tn aadrt hc is which rate hazard otone , 1 = 2 τ x icws model piecewise i n, , . . . , 0 ; ≤ i n, , . . . , . 1 = τ eae distributions Related 3          i ; x so neet see interest, of is , i , 2 1 = odel. n n , . . . , EIBULL − , 2 1 n. , . . . , dpoints nd scon- is , partition ; (3.60b) (3.60d) (3.59c) (3.60a) (3.60c) (3.60e) com- Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ofieoto h ee aaeesaeindependent. are parameters seven the of out five so hnw att aeacmoieW composite a have to want we When n otnosy ehv oitoueascn otniyc continuity second a introduce to have we continuously, ing i.31 hw h D,D n Rtgte ihteW the with together HR and DF CDF, the shows 3/11 Fig. 36a h ee aaeescmn pa knot at up too coming parameters rate, seven hazard the continuous (3.62a) a guarantee (3.62a) and (3.61a) he–odcmoiemdlwt h olwn aaee se parameter following the with model composite three–fold efis oka opst W composite a at look first We 1 with W the of Modifications 3.3 ously: st ufil(.1) hs h eutn oainparamete location resulting the Thus, (3.61a). fulfill to as omnwyt osrc 36a st xtesxparameters six the fix to is (3.60a) construct to way common the by governed is aaeesaeivle:treprmtr o ahdistrib each τ for parameters the three involved: and are DFs parameters the (see that guarantee jump not a does (3.61a) of Fulfillment i , eas ftersrcin(.1)sxo h ee paramet seven the of six (3.61a) restriction the of Because . 2 n , . . . , f i ( − a a a τ 3 2 1 i i.3/11 Fig. 1) | 1 = = 0; = a F n osittergthn D ihindex with CDF right–hand the shift to and i a i − b , ( i . τ +1 i 270987; 0 i i t W –th b c , . | 586133; 1 a a = .A hs npit,weetoCF hav CDFs two where endpoints, those At ). i i 1; = = ) i +1 a b , τ i EIBULL i EIBULL i c , − f = = b c i i +1 3 b 1 = ) b i 0 = 2 +1 0 = F EIBULL ( τ τ τ distribution 2; = 0 i i i ( F  distribution. − − . | · . 5; i 5; 0 = ) a τ +1 b

i i EIBULL +1 i c c − τ +1 b ( c 3 2 1 τ i oe eurn nyta h Dsji continu- join CDFs the that only requiring model i b , c i +1 a = 2 = 1; = 0 = i  | i and i a +1  τ i b c i − . i i c oe ihbt h DsadteDsjoin- DFs the and CDFs the both with model c , b 227= 127217 /c − b i i 1 τ i /c F i b , +1 ⇒ 2 i aet ufiltefloigtoequations: two following the fulfill to have i +1 +1 a n i +1 1 = i i ( +1  · ); ; P 1 = ) ! c i c , 3 . i c i 133= 819333 /c 1 = i 0 = +1 i 1 = +1 i ⇒ r +1 to n h atto parameter partition the and ution , / a ( , nodrt aif 36a and (3.61a) satisfy to order In . ); . i c i , 2 , 3 t: EIBULL r r needn.Temost The independent. are ers +1 i τ P 1 + 2 − n , . . . , . i i niinaogwt (3.61a): with along ondition n , . . . , 1 a , c 1 = i sgvnby given is 0 = +1 R onwtotmaking without join HRs .. odetermine to i.e., , ⇒ i b , ) n. , . . . , , . ob ond seven joined, be to e 3; P − i rbblt ltfra for plot probability c , − 2 1 0 = 1 i . b , . i . +1 4; c , i +1 a (3.62b) (3.61b) (3.62c) (3.62a) (3.61a) i +1 ( i 147 so = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC iue3/11: Figure 148 i.3/12 Fig. h portions The h loih yields algorithm The omnpoeuet osrc opst W composite a construct to procedure common A h knots the W EIBULL .Given 1. .Frtefloigknots following the For 2. Then acltdas calculated step), in r eemnduig(3.62b,c). using determined are tions τ i oe.Tepeeemndprmtr are parameters predetermined The model. ie neapefratrefl opst n com and composite three–fold a for example an gives a n h hp parameters shape the and b τ τ opst W Composite i P i +1 1 1 c , i b , 1; = ftetrescinldsrbtosare distributions three–sectional the of olw rm(3.62c). from follows i 1 and c , b 1 a 1 b , c P 1 i 1; = +1 1 0; = b 2 i c , ≈ EIBULL +1 r known. are 0 2 c ( = h oainparameters location the , . a 1 i 6321; 2 2 = 0 = c = i c +1 itiuinwt otnosyjiigCDFs joining continuously with distribution i n , . . . , . − 5; c P b 1; i i c 2 b r xd hn h loih sa follows: as is algorithm the Then, fixed. are i b /c 2 ≈ b i +1 3 i 2; = +1 − 0 ≈ h cl aaee fdistribution of parameter scale the , . 1448; 1) 7 τ . i c 862224; EIBULL h parameters the − 2 1 = a P i
3 . − a 0; ≈ ( 1 itiuinspoe hta least at that supposes distribution c a i 0 and τ − 3 . 2 2231 c − ≈ i 2 = +1 a ) 2 τ /c . i 7 . ftefis w distribu- two first the of a , 0; i . +1 eae distributions Related 3 i c . fo h preceding the (from 2 ltl continuous pletely 3 = . i (3.62d) 1 + is , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC itiuin h rpia prahrssete pnthe upon either rests approach graphical The distribution. oet fcmoieW composite of Moments hr r anytoapoce oetmt h aaeeso parameters the estimate to approaches two mainly are ca There (3.60a) because calculated easily are Percentiles tion. see h iigo eea W several of mixing The ... ie W Mixed 3.3.6.4 (1976). iue3/12: Figure W the of Modifications 3.3 aadrt,btti ilnvrhv ahu–hp swill as bathtub–shape a have never will this but rate, hazard ilas oeu nteeprclpos h ueia appro numerical The plots. empirical see the estimation; in likelihood mum up come also will E LANDT 21 K ugse edn o hsscin ic h rtppro m on paper first the Since section: this for reading Suggested vriwo hsrltv otecasclW classical the to relative this of overview ieaueo hstpchsgona nicesn ae M pace. increasing an at grown has topic this on literature AO –J M or (1959) OHNS opst W Composite HRs EIBULL /J OHNS URTHY EIBULL EIBULL EIBULL EIBULL distributions /X i.3/11 (Fig. kinks the to adheres and (1980), A itiuin saohrmast riea non–monotone a at arrive to means another is distributions distribution IE ROIAN itiuinhv ob eemndb ueia integra- numerical by determined be to have distribution /J itiuinwt otnosyjiigCF,Dsand DFs CDFs, joining continuously with distribution IANG EIBULL /R 21 20) ro h W the on or (2004), OBINS distribution. URTHY 16)and (1966) xdW ixed eivre ncoe form. closed in inverted be n /X IE EIBULL esonfrhrdw.A down. further shown be W /J IANG EIBULL opst W composite a f EIBULL c osssi maxi- a in consists ach C oesb K by models OLVERT 20)gv nup–to–date an give (2004) probability–plot, aadpo,see hazard–plot, n /2 that 3/12) and /B AO OARDMN 15) the (1959), EIBULL 149 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ie by given where ie a moments raw Mixed n r hsvrigwith varying thus are and discrete 150 in ht—cnrr otecmoiemdlo h preceding the of model composite support. the to contrary — that tions ie D n CCDF und CDF Mixed ie HR Mixed eoetrigt W to turning Before DF has tion ie DF Mixed butions: qual continuous some that so identically perfectly working ouainudrcnieaini aeu of up made is consideration under population o fiesdlvrdb atr a enpoue on produced been has factory a by delivered items of lot a 23 22 ω 92 namxueo nivreG inverse an of mixture a on 1992) ewl o present not will We wfl itr fW of mixture twofold a aees ilb rsne nSc.3365 nti aeth case this In 3.3.6.5. Sect. in presented be will rameters, namxueo W of mixture a on A W L i ANDES h i , EIBULL continuous F 23 i ( or i x ( 1 = x = ) ipeepaaino iceeymxddsrbtosi t is distributions mixed discretely of explanation simple A nt mixture finite ) 19)o itr fnra n W and normal of mixture a on (1993) aiy oeppr nhbi itrsaeA are mixtures hybrid on papers Some family. hnpoue nline on produced when , 2 f E n , . . . , i or ( x X nnt mixture infinite ) f EIBULL EIBULL /R r m
yrdmixtures hybrid ( 0 ; = i x EIBULL ( = ) x x X i ω < 22 ) ; =1 g n n G and itrsw umrz oegnrlrslso ie distri- mixed on results general some summarize we mixtures stehzr aeo subpopulation of rate hazard the is i ( salna obnto ftoo oeW more or two of combination linear a is X i h ω x =1 i n n nfr itiuin o rdcigatmbl warra automobile predicting for distributions uniform and F m = ) i R < OMPERTZ AUSSIAN m E hr h egtn ucini Ffroeo h W the of one for DF a is function weighting the where , ( ω m x ( 1; i X x x = ) ω f = ) = ) R P i i i i r hr oeo h ie itiuin ontbln othe to belong not do distributions mixed the of some where . ( =
R m x i n , =1 n w–aaee W two–parameter a and ( ;0 ); i itiuin omdltedahrt nhmnlife–tables, human in rate death the model to distributions ( hr E where x x X ) i R ω f EIBULL =1 ) n X X i i m i m , =1 =1 n n ω < n 1; = ( ( g x x i X ω ω i ≥ ( ) =1 ) n x i i i itiuin n M and distributions X F R 2 ) < oteetr ouain o example, For population. entire the to W e L g i h i i uppltoscnrbtn h por- the contributing subpopulations r –H i ( ( i ( 1;
x x ( x EIBULL ); x USSAINI = ); 1 = ) ) n X t characteristic ity i , =1 Z n rdcinlnswihaenot are which lines production EIBULL eto aen restricted no have — section x ω aaee sarno variable. random a is parameter i r i /A i f n h weights the and 1; = BD i ( itiuin C distribution, AJESKE x – eae distributions Related 3 ) EL d efloig The following: he –H x. EIBULL /H AKIM X ERRIN fteitems the of t claims. nty HANG 18,1990, (1989, EIBULL g distribu- 19)on (1995) i (3.63d) (3.63b) (3.63e) (3.63a) (3.63c) ( (3.63f) x (1998) ) are pa- Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ehave we and esatb rtdvlpn oeapoiain oteCFof CDF the to approximations some developing first by start We hr the where Denoting n ntesqe,wtotls fgnrlt,that generality, of loss without sequel, the in ing no(.3–)gvsteseilrslsframxdW mixed a for results special the gives (3.63a–h) into netn h omlsdsrbn h ucin n moment and functions the describing formulas the Inserting siain(e ate on h eairo h est an density the of behavior the down) farther (see estimation ayaon hi omnma E mean common their around vary variance emdthe termed h rtcmoeto h ih–adsd f(.3)i h me the is (3.63h) two of of side sum right–hand the the on is component distribution first mixed The the of variance the Thus, eno h ie distribution mixed the of Mean results: special two have we (3.63f) From W the of Modifications 3.3 aineo h ie distribution mixed the of Variance W 24 EIBULL uinrdcst w–aaee itiuinudrashif a under distribution two–parameter a to reduces bution h eut o he–aaee W three–parameter a for results The or n itiuinhv trce h teto fstatistician of attention the attracted have distribution nenlvariance internal uppltosalhv w–aaee W two–parameter a have all subpopulations between–variance Var ( X x = ) lim →∞ x = = = lim → EIBULL  0 y X X X E y  i i i =1 =1 =1 1 n n n i  or y y  1 i hwn o h en fteidvda distributions individual the of means the how showing , ω ω ω X within–variance distribution  = E i i i ( Var EIBULL (  E − X =      X Var E X ) ( = )      y . X (  ( i i 2 X X
i = b b  + ) itiuinaesmlrbcuetetreprmtrdist three–parameter the because similar are distribution ) 1 X i i − i  + ) =1 n b  b  ∞ 2 0 1 i c "

1 X b i  x ω X =1 i n i E c =1 = > n i c  1 ( i E c X ω h eodcmoeti the is component second The . E 1 i ≤ ω i if if i X i  ) X c igo h iescale. time the of ting E  E if if i j 2
EIBULL 2 ( c c j, < i , ; X X
− i i EIBULL c c = c > − i i i i "
) of s # = c > X  c i − h aadrt famixed a of rate hazard the d =1 2 E n 1 1 c ( E o–eaiecomponents. non–negative oe.Bsdsparameter Besides model. and s. no the of an      1 1 X ω ie W mixed a n ( distribution X i      ) W E  ) b 2 ( .  i EIBULL X 2 b > . i ) n # j 2 when , aine.I is It variances. EIBULL 24 distributions n assum- and external c (3.64b) (3.63h) (3.63g) (3.64a) i model = 151 c ri- j . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC cl aaeesadtemxn aaee.J parameter. mixing the and parameters scale 1 sh possible The mixture. two–fold the is interest special Of lhuhtetofl itr oe a v parameters five has model mixture two–fold the Although of rm(.6,)the (3.66a,c) four–dime From this in function density the of characterization mligthat implying and and rm(.5,)the (3.65a,b) From rm(.4,)J (3.64a,b) From 152 o ftehzr n est ucin a ededuced: be may functions density and hazard the of ior − f .Frlarge For 2. .Frsmall For 1. • • • • • • m ω ( 1 bimodal. unimodal, by followed decreasing unimodal, decreasing, k eraigfloe by followed decreasing When and a eapoiae by approximated be can where x h Fsaei nyafnto ftetosaeparameters, shape two the of function a only is DF–shape the , –modal ) eed ntemdlprmtr,adtepsil hpsare shapes possible the and parameters, model the on depends m m stenme ftesbouain ihtecmo hp par shape common the with subpopulations the of number the is f m 1 = ( x x IANG k ( , .. eycoet zero, to close very i.e., , x F 1 = aadrt ftemxdW mixed the of rate hazard est ftemxdW mixed the of density ) then , m sicesn dcesn)frsmall for (decreasing) increasing is ( /M , x 2 ) f n , . . . , URTHY f g h m a eapoiae by approximated be can h m m = ( m F ( x ( x m ( k ) x ω x ) ) ( ≈ 1 − ) ) x ≈ F . . 19)drv h olwn eut rmwihtebehav- the which from results following the derive (1995) ≈ ≈ ) m F ω 1 f g ≈ h g m 1 ( modes h x f ( 1 1 1 = ) g x 1 1 ( ( ( ( x − ) x = x x ≈ | | ω | X | i 0 X 0 j ( =1 0 n 0 m EIBULL =1 1 b , b , F g k b , b , h 1 1 1 = ω 1 1 ω 1 c , c , 1 IANG c , i c , − j EIBULL ( F 1 x 1  1 , 1 ) ) i F ) 2 | ) ( b b o large for 1 0 n , . . . , x model o small for 1 /M j o small for o large for ( b ,  | x 0 c 1 URTHY | soa aaee space. parameter nsional 1 b , c , x model 0 ( b , psi hscs are case this in apes b i 1 if c , a eapoiae by approximated be can 1 − ) 1 b , , x. c i c , 1 1) x ) x x, 2 1 a eapoiae by approximated be can > 19)gv parametric a give (1998) , c , ) i ( 1 . 1 c , eae distributions Related 3 c h ai ftetwo the of ratio the 1 2 < and 1) ω h shape The . 1 ameter ,as ), (3.66b) (3.67b) (3.65b) (3.66a) (3.67a) (3.65c) (3.65a) ω 2 c = 1 . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h odtoso h parameters the on conditions The are (3.68a) of shapes possible The sdcesn icesn)frsaladlarge and small for (increasing) decreasing is When nJ in 36bc ml htfrsaladlarge and small for that imply (3.68b,c) Assuming orsodn est ucin(ethn ie o h fiv the for side) (left–hand function density corresponding with h Thus, W the of Modifications 3.3 pca aeo eraigmxdH a led engiven been already has HR mixed decreasing a of case special A epoe htamxueo xoeta itiuin,bein distributions, exponential of mixture a that proved He c enwtr otehzr aeo w–odW two–fold a of rate hazard the to turn now We rate. hazard decreasing with model oedti.Ti aadrt olw rm(.3,)as (3.63d,e) from follows rate hazard This detail. some G h i m m UPTA 26 25 1; = ( ( • • • • • IANG • • • • x x hp;tu asfigteasrino K of assertion the falsifying thus shape; having n a vnso that show even can One ae ihnwrrslso itrso xoeta distri exponential of mixtures on results newer with paper A ) )= imdlfloe yincreasing. by followed bi–modal uni–modal, by followed decreasing increasing, by followed uni–modal increasing, decreasing, eraigfloe by followed decreasing increasing, decreasing, k c h /G 1 eed ntemdlprmtr edn otefloigpos following the to leading parameters model the on depends m oa olwdb increasing by followed modal /M i < ω ( UPTA c x c 1 1 = g 1 1 ( 1 ) URTHY R = < ≤ sicesn dcesn)frsmall for (decreasing) increasing is 1 c R , 1 ( 1 ω c (1996): 2 x c , 2 m 1 > n , . . . , | J 2 ( 0 for IANG x 1) (1998). 1 = b , ) h then , 1 c m c , h 1 and hc aedfeigsaeparameters scale differing have which , EIBULL ( n /M m c < 1 x fl itrso W of mixtures –fold ) ( ) c x h i.3/13 Fig. URTHY 3 h → ) 2 1 k > 1 ( ( → x modes and x 1 h g distribution | ed oabttbshape. bathtub a to leads . b | h ) 1 0 1 1 b , sdcesn icesn)frall for (increasing) decreasing is b , 19)show (1998) ( g ROHN ( x x hw aadrt rgthn ie alon side) (right–hand rate hazard a shows ( 1 = ( 2 k | x c , | k c , 0 0 1 = ω h hp of shape the b , 1 1 = b , 16)ta he–odmxueo W of mixture three–fold a that (1969) + ) 1 1 EIBULL 1 c , 1 (1 + c , , c , , x 2 2 EIBULL 2 1 so , (1 n , . . . , 1 n , . . . , ) ) and − − o small for distributions o large for x h ω ω m ω if 1 uin sJ is butions 1 h ) edn oteesae,aegiven are shapes, these to leading , ( ) ) itr,wihhsbe tde in studied been has which mixture, . m c x − R  R 1 ) osbehzr shapes. hazard possible e ( m 1) 2 x b b anthv ahu shape. bathtub a have cannot > ( 2 1 W g ( ) , x x, x, x  ( ssmlrt htof that to similar is > n ) ( 1 | c 1 EWELL 0 EIBULL b , yP by c b for 1 2) 2 i illa oamixed a to lead will , c , x < a ee aeabathtub a have never can ROSCHAN (1982). 2 saresult, a As . c ) 1 1) il hps see shapes; sible EIBULL h = itiuinwith distribution h hp of shape The . 2 ( c x 2 . | 0 distributions ihits with g b , (1963). h 1 (3.68b) 2 (3.68a) (3.68c) h ( c , x m 2 | 153 ( ) . x 26 25 . ) ) . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC nysm e papers. key some only edmnn.Tefis ae ae nti prahi K is approach this on based paper first The dominant. be h ieaueon literature The ee ae icmetn oeo h rwak fodrapp older of drawbacks the intro of authors some several circumventing by paper applied newer been has approach His ture. 19) h rpia ehdmyla ostsatr resul satisfactory to i.e., lead cases, may well–separated method graphical The (1995). aaee siainare estimation parameter iue3/13: Figure 154 • • • • • odapproach, hood R C W A A IDER HENG SHOUR SHOUR OODWARD aadrt n est ucino oetofl W two-fold some of function density and rate Hazard 16)adF and (1961) /F 18b n A and (1987b) 18a n J and (1987a) U aaee estimation parameter 18)cosn egtdlatsursestimator, least–squares weighted a choosing (1982) /G UNST rpia methods Graphical ALLS b 1 18)tkn iiu–itneestimator, minimum–distance a taking (1987) ≫ IANG HMAD 17)uigtemto fmoments, of method the using (1970) b 2 /K or ta.(97 osdrn B a considering (1997) al. et ECECIOGLU b 1 o h itr oe svs.S,w ilcite will we So, vast. is model mixture the for ≪ ae h W the based b 2 oeppr on papers Some . 19b pligtemxmmlikeli- maximum the applying (1992b) EIBULL AO uigmnrmdfiain.A modifications. minor ducing sfratofl itr in mixture two–fold a for ts oce sJ is roaches 15)o w–odmix- two–fold a on (1959) rbblt ltse to seem plot probability AYES EIBULL nltclmethods analytical eae distributions Related 3 approach. IANG mixtures /M URTHY of Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC opudW Compound eral h eut etiigt hscniinlW conditional this to pertaining results The o cl factor scale dom yD by nrdcdb .W W. by introduced sreproductive, is hs h aetlW parental the Thus, V No σ oaini sdt xrs opudn.Freape e th let example, For mean compounding. random an express with statistics normal In to distribution. used compound is the notation of DF the is This rmwihthe which from The termed opudn ti termed is it compounding Θ aeweeoeo h aaees eoe by denoted parameters, the of one where case termed is mixture uous aaee faprmti itiuinmyb eadda c a as regarded be may probabilities distribution parametric a of parameter stepoaiiisof probabilities the as weights The W Compound 3.3.6.5 W the of Modifications 3.3 ∗ 2 h est of density The . 27 28 i.e., , sthe is ( ,σ ξ, hoe fE of theorem on density joint ugse edn o hsscin D section: this for reading Suggested h term The S UBEY INGPUWALLA opudn distribution compounding ∗∗ 2 X opudn operator compounding ) opudn nti aei eoe by denoted is case this in compounding , | 16)udH und (1968) θ ro distribution prior ω ω EIBULL i ∼ i sdi mixing in used , agnldensity marginal aet esbtttdaantadniyfnto.Teresul The function. density a against substituted be to have hssaefco stercpoa ftecmie scale–s combined the of reciprocal the is factor scale This . No LANDT ftevariates the of 16,1969). (1968, X EIBULL ( EIBULL f given , ,σ θ, f No X itiuin aebe nesvl icse ihrespec with discussed intensively been have distributions n ( ( opuddistribution compound EIBULL x eso ausfrteprmtrvector parameter the for values of sets x (Θ EIBULL ARRIS –J ∗ 2 = ) aetldistribution parental ) | Θ sue nB in used is upsn ro itiuin hc sas normal, also is which distribution, prior a Supposing . β σ , e (2.26a,b): see ; OHNSON θ , Frasas reads DF = ) scniinladdntdby denoted and conditional is , Z θ ∗ 2 f ) /S No n en elzto of realization a being distribution X ( f distributions esotymninta nti xml compounding example this in that mention shortly We . V Θ β c ,θ x, INGPURWALLA of W ( (Θ ,θ x, or and β No X EIBULL ( AYES = ) σ , x (1976): =1 := ro distribution prior ) ( olw as follows Θ − ∗ 2 ,σ ξ, d ) UBEY θ f neec,seSc.1,wihi eae ocompounding. to related is which 14, Sect. see inference, sgvnby given is a /b X V Θ EIBULL = ) ∗∗ 2 c ( c densities − x = ) No Z 27 = 1 | 16) E (1968), Θ h est of density The . ntesqe ewl ics nythe only discuss will we sequel the In . θ exp f b ( ) 16)aeseilcsso oegen- more a of cases special are (1968) srno.Let random. is , ,σ ξ, − No X f c itiuinwihhv enreported been have which distribution ( Θ . {− ,θ x, ( ∗∗ 2 Θ ( ,σ ξ, f θ i β ) n xdvrac Var variance fixed and , ) ( . . ) LANDT , 28 x ( f ∗ 2 x Θ | + rbblt hoyaspecial a theory probability d a ( − niuu ait,s htthe that so variate, ontinuous θ i f σ b , ) X –J a ∗∗ 2 aetldsrbto be distribution parental e d ( ) i ( OHNSON ,b c b, a, Θ c , θ. c ) x } . eoe by denoted , i | . ) θ θ a einterpreted be may , ) earaiainof realization a be ) ntecnetof context the In . eeal,each Generally, . 17) H (1976), aeparameter hape igcontin- ting oa to t ( f X Θ (3.70b) (3.69b) (3.70a) (3.69a) ( | ARRIS θ Θ θ ran- ) = ) 155 is , ∼ / Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC no(.1)yed h olwn D ftecmon W compound the of CDF following the yields (3.71b) into Putting h nfr rrcaglrdsrbto over distribution rectangular or uniform The eoe by denoted h orsodn a oetgnrtn ucinis function generating moment raw corresponding The Teprna W parental (The ( rvddthat provided ewl pl hstermt aetlW by parental denoted factor, a scale to stochastic theorem this apply will We ro.Teepirdsrbtosaetems oua nsi case: ones First popular most the are distributions prior These prior. x hn h D ftecmon itiuini ie by given is distribution compound the of CDF the Then, eteCFo h aetldsrbto,wihi of is which and distribution, parameter parental the of CDF the be Let 156 W x with − EIBULL a F ) ( c x )Frhr e h opudn itiuinhv a mome raw a have distribution compounding the let Further, .) | ,δ ,c ξ, δ, a, uiomdistribution –uniform We M α F Λ ( X EIBULL savco f“riay parameters. “ordinary” of vector a as a,  ( 1 = ) F − x, p c ( u x α 1 ( = ) B c /B, f x, | − B λ itiuin(.0)i fti yewith type this of is (3.70b) distribution α ( = ) M exp    y ) u u ) = )  B t ( ( V B 1 xssfor exists    x, x, (  = t − = ) Re    − ): α α B − 1 E M ) ) a ihrauiom( etnua)piro gamma a or prior rectangular) (= uniform a either has , ξ − u ( Λ 1 ,δ ξ, B Λ ( ( exp( /δ 0 →∞ −→ −→ ( ,a c a, x, x exp e  λ t ∼ t δ − − ) x > x 0 =: ) t ξ {− for for Re u a 0 EIBULL ) [ ( = ) 0 ) ,ξ ξ, c x, u λ (

 ,δ ξ, 0 M δ ξ ξ as as . α exp( − ( − ( + elsewhere ≤ Λ x ) x, ) ( exp  xoeta type exponential ( . δ x − itiuino ye(.0)weethe where (3.70b) type of distribution t y x x α t δ ] ) − . a DF has for for ≤ a ) ∞ → → u  } ) ) ( a c x, − ξ − ) x B n c + for for 0 α 1 x > x x , ( . , ξ  δ ) AYES ≤ . + sa nraigfnto of function increasing an is    x > x x x δ EIBULL 0 0 ( ) ≤ λ , inference. tgnrtn function generating nt x    eae distributions Related 3 x = with 0 0 − β    a itiuin(or distribution ) λ and c

sa“special” a as u ( x, (3.72b) (3.71b) (3.72c) (3.72a) (3.71a) α = )  Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC (or ihDF with eodcase Second nteseilcase special the In n(.3)trs(.1)it h olwn D fti compo this of CDF following the into (3.71b) turns (3.73b) in aiirrslso h omntreprmtrdistributi three–parameter common the of results familiar Taking hntepirdniy(.2)degenerates (3.72a) density prior the When ihsm nerto n aiuainw find we manipulation and integration some With ihrwmmn eeaigfunction generating moment raw with h he–aaee am prior gamma three–parameter The ihcrepnigDF corresponding with W the of Modifications 3.3 f exp ( x W f | (  ,δ ,c ξ, δ, a, EIBULL x − Var | ( ,δ ,γ c γ, ξ, δ, a, ξ ( + X : = ) We gmadistribution –gamma = ) δ E )( ( X ( x a, δ c ξ = ) − = ) E × F 0 = p ( c a δ δ X ( f 2 ) 1 x a ( c δ c B hr xs lsdfr xrsin o h enadvarian and mean the for expressions form closed exist there , B c /B, c ( | EIBULL  | + c ξ ( c − ,δ ,γ c γ, ξ, δ, a, c y γ − 2 )= 0) = exp ( δ = ) 1) x 1) ) t ( c − = Γ V  B 2 c − δ distribution M δ  a γ Γ − Ga Ga ) ( 1) B + 1 2 c x a u y ): −  1 = ) ( Γ( ( ( ( ( + − t Γ − 1 + 1 x ,δ γ δ, ξ, ,a c a, x, ,δ γ δ, ξ, = )   γ 2 c a − δ ξ γ )  + 1 ) 1
c /c − γ a 1 + c h δ − ) ( ) γ  ) = ) (  Γ c ξ ( 1 ξ δ δ 1 c δ +1 + 1 over δ h  exp γ +   ( exp ( + − δ γ ξ exp → −  h δ ( + δ +  − ξ ( t ) ( [ x ξ )  1 x t ξ ξ, ( + ( Γ( γ  − δ x 1 + c − 0) x − − ) 1 ∞

γ  − 1 − /c − x h omls(.2–)so the show (3.72c–f) formulas the , δ a − δ ) . a ) nfor on ) ) ξ Γ − 1 a ) − ( 1 a c a DF has /c c y ( ) −   ) x c a γ ξ c 1 −
− 1 c +1  ) /c 1 − − c − n W und ξ 1 +  ξ b 1 − ( γ ) a exp 1 /c ξ

= − ) ξ c i  + 1

1 ξ /c  − − − δ , EIBULL i 1 − )( 1 /c 2 /c x . ξ i . (          , − x a − ) distribution c a − ) c 1 (3.73d) (3.73b) (3.72d) (3.73e) (3.73c) (3.72e) (3.73a) (3.72f)

 ce: . .        157 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC efrhrmninta nteseilcase special the in that mention further We 16)we aaeeshv ob siae o aho h tw the of each for estimated be to have parameters when (1969) onatrteW the after Soon W 3.3.7 distributions. ntepeeigscin,w aearayecutrdseve encountered already have fr we sections, parameters preceding these the identify In properly hav not and to involved risk are the that be parameters will more flexi the more But, at aimed datasets. authors those parameters, additional ing nterpprfo 97 M 1997, from paper their In ... orprmtrdistributions Four-parameter 3.3.7.1 set. parameter oeta h rdtoa oain cl n hp paramet shape and scale location, traditional the than more D n CF epciey h CFo h e distribution new the of CCDF The respectively. CCDF, and CDF oitouea diinlprmtrit ie distribut given a into parameter additional an introduce to nti eto ewl rsn ute W further present will we section this In Substituting itiuin earv ttefloigCD fthe of CCDF following the at arrive we distribution, 158 B W M URR EIBULL ARSHALL • • • • Var h obytuctdW truncated doubly the h etadtergttuctdW truncated right the and left the h utpiaieW multiplicative the h W the each, tdW ated yeXIdsrbto;se(.ab.Fnly erfrto refer we Finally, (3.3a,b). see distribution; type–XII ( X EIBULL n15,saitcasbgnt eprmtrz hsdistr this re–parameterize to began statisticians 1951, in | EIBULL ξ /O F EIBULL )= 0) = ( LKIN x G ) R EIBULL ( and x ( itiuin ihadtoa parameters additional with distributions optn ikmdl(.7,)wt orparameters, four with (3.57a,b) model risk competing x = ) oe,ec aigfu parameters. four having each model, extension ( Γ | δ R ,b ,α c, b, a, 2 /c γ ( 1 x EIBULL ) − itiuinhdbe rsne oabodrraesi by readership broader a to presented been had distribution ) ARSHALL  EIBULL yteCFadCD ftetreprmtrW three–parameter the of CCDF and CDF the by (1 Γ R α = )  − γ ( α oe 35ab n t eeaiain h exponenti- the generalization, its and (3.58a,b) model x − ) 1 ) itiuin(.7)wt v parameters, five with (3.47a) distribution /O R EIBULL − 2 c ( EIBULL  LKIN x (1 α ) Γ − = exp  ξ α rpsdanwadrte eea method general rather and new a proposed 2 itiuin 34bc ihfu parameters four with (3.47b,c) distributions c F exp ) 0 =  itiuin,elre ihrsett the to respect with enlarged distributions, 1 + ( x − R α + ) xeddW extended   and  x − ( − R α x − b Γ( )  mtedata. the om δ r,namely: ers, a W ral o having ion a ( 1 iiyt ttemdlt given to model the fit to bility x γ x ob siae,tegreater the estimated, be to e 1 =  ) c ) − b  Γ H > α , EIBULL a sgvnby given is 2 EIBULL 37cd unit the into turn (3.73c,d) , ARIS opudW compound o   c bto.B introduc- By ibution. 1 c  eae distributions Related 3 1 + . F 0 /S . itiuin with distributions ( distribution x  INGPURWAL ) and . R EIBULL EIBULL (3.75a) (3.73f) ( (3.74) x ) : as Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ihDF with n HR and h aadrt ie y(.5)hstefloigbhvo f behavior following the has (3.75d) by given rate hazard The hycall they a 1 CF(.4 and (3.74) CCDF ehave we When h orsodn D is CDF corresponding The W the of Modifications 3.3 M L L , AI AI 0 = 29 30 2 ARSHALL . . . , • • • • ta.(03 rpsdtefloigetnino h W the of extension following the proposed (2003) al. et extension al. et narcn ril N article recent a In yG by N see censoring, type–II under estimation parameter For for for for decreasing, is it where interval for nevlweei sincreasing, is it where interval n sn h obndsaesaeparameter scale–shape combined the using and | 1 ; − URVICH hntemnmm(aiu)of (maximum) minimum the then emti–xrm stability geometric–extreme α α < c > c F α ≥ ≤ ( /O x ≤ | 1 1 1 1 LKIN | 0 ta.(97;sealso see (1997); al. et and and ti ntal eraigadeetal eraig u t but decreasing, eventually and decreasing initially is it f ti ntal nraigadeetal nraig u t but increasing, eventually and increasing initially is it b , 1 ( h N oet fti itiuincnb ie ncoe om Pa form. closed in given be can distribution this of moments , F x 1 ( ,α c, , c c ( rvdta h itiuinfml eeae y(.4 pos (3.74) by generated family distribution the that proved x | ADARAJAH x sgoerclydsrbtd ..Pr i.e. distributed, geometrically is ,b ,α c, b, a, ≥ ≤ | E | ,b ,α c, b, a, ,b ,α c, b, a,  1 1 EIBULL 1 = ) ( ti increasing, is it ti decreasing, is it X − = ) /K − = ) a distribution = ) OTZ ) exp L r c α   AI b 1 1 20)nt htti xeso a enmd oeyasearl years some made been has extension this that note (2005) = 1 {− ta.(2005). al. et − − If :  − c b r b (1 (1 x (1 1 1 X − X b x − X − − j − ∞ =0 c i i a i , c b exp α α e lohsadsrbto ntesm family. same the in distribution a has also α  x α exp ) exp )  1+ (1 c exp ) (1 − ≤  x } 1 G − − ; − b N exp (2005). j   30 α  b  ) a sasqec fidvrae with variates iid of sequence a is , 1 − − r/c )  − x b  > c , j ( c 1   N − − −  b Γ 1 = 1 x x  x  a rgrowing or = − − b EIBULL b r  c 0; − x b /b c  a a n  − a b c α .   = ) :  c c a ≥ c     c 0 . model,  P 2 . x eemyb one be may here eemyb one be may here (1 . − 29 esswhat sesses P assuming rticularly ) n (3.75b) (3.75d) (3.75c) (3.75e) (3.76a) ; n 159 ier = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h ML–method. the efial eto htL that mention finally We h eairo h aadrt 37c so pca interes special of is (3.76c) rate hazard the of behavior The n h w te parameters other two the and edsigihd(see distinguished be Percentiles ehv h olwn eainhp ihohrdistributio other with relationships following the have We oet fti xeddW extended this of Moments h orsodn F RadCRare CHR and HR DF, corresponding The 160 • • • • • hc sicesn when increasing is which Ri increasing. is HR c h 0 h x α 37a salmtn aeo h eaitgae oe with model beta–integrated the of case limiting a is (3.76a) distribution. log–gamma a as and c exp lim → (0 ( ≥ 0 = < c < 0 = x 0 { | > g | 1 h x α · x · ( ) = ) H eut ntetp– xrm au itiuinfrtemin the for distribution value extreme type–I the in results P x f h ie h eea W general the gives siiilydcesn n hnicesn,ie thsaba a has it i.e. increasing, then and decreasing initially is } | 0 ( ( ( 1 forder of x x x · and , < k , b lim = ) | | | 1 0 0 0 c b , b , b , for x H i.3/14 Fig. →∞ 1 1 1 0 ,α c, , ,α c, , ,α c, , Set . ( P c x x  ) AI 1 = olwb solving by follow h h P H stesm s(3.76d). as same the is ( ( = ) = ) = ) . g x x = ta.(03 hwhwt siaetemdlprmtr by parameters model the estimate to how show (2003) al. et EIBULL α ( . ): 1 = | |  x  · · α and = ) ) = ) − EIBULL decreases. /ν ln(1 ∞ → b b b b b 1 1 1 ∞ 1 and 1 oe aet eeautdb uei integration. numeric by evaluated be to have model x ( ( x x aen fet ihrsetto respect With effect. no have b c c . c − 1 h, c h iiu of minimum The + + exp k distribution. (1 min P as x α x α = ) − { = x ν α x α exp ) ) x g x x √ c c ∞ → With . } {− ) − − c . k α 1 1 − 0 ; x α exp exp c ν , but , g < x < P ns: {  h .Issaesll eed on depends solely shape Its t. } ∞ → x α x α (  x 1 /c h } | − · (0 , . ) ehave we − b CHR srahdat reached is | 1 1 · ; 0 = ) x eae distributions Related 3 c c e mm loknown also imum, x α w ae aeto have cases two hu hp with shape thtub (1

for − , x/ν > c ) − (3.76d) (3.76b) (3.76e) (3.76c) (3.76f) ν α 1 and → c , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ute parameters: further h upr f(.8)dpnsntol on only not depends (3.78b) of support The h ii fteratio the of limit The iia otedfiiino the of definition the to similar fatm eis o for So, series. time a of P iue3/14: Figure W the of Modifications 3.3 F oiyn hsfnto n nrdcn orhparameter fourth a introducing and function this Modifying se(2.57b)): (see h ecniefnto ftegnrltreprmtrW three–parameter general the of function percentile The function percentile the modifying by Extensions prahsaogti line. this along approaches M ( UDHOLKAR = x | F . . . ( x ) | saohrwyt riea xeddW extended at arrive to way another is · ) aadrt fteL the of rate Hazard gives ta.(96 rnfre 37)into (3.77) transformed (1996) al. et F [1 ( x x − EIBULL λ P | (1 ,b ,λ c, b, a, = 0 = x − a B P + OX P distribution ehv h eea W general the have we = AI ) b 1 = ) λ –C a ta.etnino h W the of extension al. et  ]  + 1 λ OX − b n(.8)for (3.78a) in −  –transformation (1 −  λ a − 1 ln(1 ntesnethat sense the in − P ) λ − λ EIBULL   P 1 /c x ) λ  − λ , b EIBULL 1 /c EIBULL → a sdt tblz h variance the stabilize to used , . oes ewl ics two discuss will We models.  EIBULL ∈ c ν 0  R 1 n eivrigto re–inverting and x is /λ itiuin ovn for Solving distribution. . itiuini ie by given is distribution ≥ − . model ln(1 a u loo some on also but , − P ) hc is which , (3.78b) (3.78a) (3.77) P 161 = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC iue3/15: Figure 3/15). Fig. (see h eairo 37d sa follows: as is (3.78d) of behavior The h FadH eogn o(.8)are (3.78b) to belonging HR and DF The 162 The • • • • • • • r t oetof moment –th c > c c < c c > λ λ ≥ ≤ 1 = ≤ f 1 1 1 1 0 0 ( and and and and and ie h support the gives ie h support the gives x model aadrt fteM the of rate Hazard | ,b ,λ c, b, a, h λ < λ λ > λ λ ( x ≥ ≤ 0 = | ( 0 0 0 0 ,b ,λ c, b, a, X = ) ⇒ ⇒ ⇒ ⇒ ⇒ − a ) b c h h h h h = ) codn oM to according  ( ( ( ( ( ( ( ,b a, a, x x x x x x | | | | | ∞ c b · · · · · − b  ) ) ) ) 1 = ) UDHOLKAR  ) λ sincreasing is sivrebathtub–shaped, inverse is sdecreasing, is sbathtub–shaped, is a . 1 x  /c c − b ) /b − . 1 a =   UDHOLKAR 1 c constant, − − 1 ta.(96 xeso fteW the of extension (1996) al. et λ  1  − x λ − b /K  a x OLLIA  c − b  − a 1+1  19)is (1994) c eae distributions Related 3  /λ − 1 , . (3.78d) (3.78c) EIBULL Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ie n(.8) 37a nldstefloigseilca special following the includes (3.79a) (3.78e). in given h upr fteels he ucin eed ntepara the on depends functions three last these of support The h a oet of moments raw The ihD n HR and DF with nlddi hsmdl ovn for Solving model. this in included hudmninta M that mention should ecniefnto,assuming function, percentile ML. if Here, 37a,cruvnigisdsotniyat discontinuity its circumventing (3.78a), ec,if Hence, . oictoso h W the of Modifications 3.3 M λ/c UDHOLKAR • • • • ∗ < c ∗ ∗ > c λ ∗ λ c − ≤ E 0 = 0 = a engtv,to ota h nes W inverse the that so too, negative, be may If If If If > λ  0 0 ( < λ > λ < λ > λ r X > c , < c , − 1 /K 0 − f h n ncs of case in and , 0 0 0 0 G and G ehave we , ehave we , ehave we , ehave we , a OLLIA ( ( 0 0 ) x x r  > c | ⇒ ⇒ X | 0 0 = F , , UDHOLKAR itiue codn o(.9) a eepesdb h m the by expressed be can (3.79b), to according distributed , G 1 1 x 19)hv tde oegnrlzdpretl function percentile generalized more a studied have (1994) eue W reduced nes eue W reduced inverse                            0 EIBULL ,λ c, , ,λ c, , ( P x oet falodr xs.For exist. orders all of moments , x x x x b b = | ∈ ∈ ∈ ∈ r r 0 a = ) = ) , c Γ Γ 1 Γ ( ( ( ( 1 0 = −∞ −∞ − −   c< /c  ,λ c, ,  distribution Γ 1 − c, ,c c, λ 1 1  P EIBULL − ta.(96 hwhwt siaeteprmtr by parameters the estimate to how show (1996) al. et and ∞ r − c , c , + λ 1 1 = )   h − = 0 − (1 1   ) λ + 1 λ 1 r c c λ e h etprgah—if — paragraph next the see — λ 1 − . b F )  λ − 1 /c Γ 1 + 1 1 = . ( /c  λ  EIBULL λ x x − P c − distribution, r  c  −  − | Γ ) 0 = h : c + 1 c · λ 1 +  ) 1 − ) c λ λ  1 ) ed to leads r c . − 1
r/c /c r/c . nta,te rpsdtefollowing the proposed they Instead, .  h x 1 + c λ distribution, 1 +1 −  +1  − c EIBULL  + 1 ,λ c, i λ λ 1 ses: −  for 1 for x + 1 c < λ meters ∈   c + 1 itiuin(et ..)is 3.3.3) (Sect. distribution R i x c < λ > λ 1 . 0 /λ  the c x c c i  − 0 0 and c . , r 1 − t oetexists moment –th . 1 1                            /c λ , sfollows: as − ≥ r − (3.79d) (3.79b) oments (3.79c) (3.79a) (3.78e) 1 We . than 163 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC hthsafiieitra ssupport: as interval finite a has that f with with h eairo Ri eemndb h hp parameter shape the by determined is HR of behavior The efidtefloigrltost te distributions: other to relations following the find We h iiigcases limiting The yitouigalcto parameter location a introducing by 164 K K h X X ( ( IES IE IE IES x x • • • • • • • • = ) = ) ta.(02 rpsdadsrbto htw aeelre t enlarged have we that distribution a proposed (2002) al. et ta.extension al. et ,α > β α, λ, ,β> β λ, 15)itoue h olwn orprmtrvrinof version four–parameter following the introduced (1958) When (1975). a eapoiae yaW a by approximated be can β For i.e., 0 When (2000). λ λ λ extension < β < 1 = 0 = = ≥ β λ β λ h α − 1 ( c , c , 0 α λ x α 1   1 = ⇒ h orsodn FadH are HR and DF corresponding The . c , ) F F 0 slre then large, is x x 1 1 = 1 = a ahu shape. bathtub a has ( 1 = and ( − − α α h = x x ⇒ ehv h rgnlvrino hsdsrbto introduce distribution this of version original the have we ( ( 1 = ) 1 = ) a a x c ⇒ ∞ hsmdli eae oteepnnilpwrmdlo S of model power exponential the to related is model this , ⇒ ⇒ a )   ∞ → h sicesn from increasing is β β ∈ ( − − x nfr itiuinover distribution uniform xoeta distribution, exponential − − 1 1 R ) exp λ , h orsodn FadH are HR and DF corresponding The . sdcesn from decreasing is exp exp oitcdistribution, logistic exp 1 ( − → ( ( (   α λ exp − 0) x x x EIBULL λ a ∗ − α r nepee codn oL H L’ to according interpreted are − ( α h D is CDF The . 

= a  x 1 a b  a  h − x − − β + ( β ) − α a distribution. x exp a ) α 0 = ) exp  a x < x . (1 β  " ) β ( /β  [0 ) if , α λ , x , ∗ − − ≈ 1] 0 − α > β

n eraigfor decreasing and 1) , ≤ 1 a 1  − a β /β exp 1 β : ≤ exp rfrom or , #!) ,  x " orprmtrmodel four–parameter a o x ≤  W a x , − α x < b a eae distributions Related 3 − α h EIBULL OPITAL  ≥ ( a ∞ a β  a, = ) x > x ota (3.80a) that so , , β #!) λ srule. ’s MITH yC by d distribution if , ∗ where , , β (3.80a) (3.81a) (3.80b) (3.80c) /B 1 = HEN AIN . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC olwn way following 38a otisalto te itiuin sseilcas special as (see distributions eters other of lot a contains (3.82a) with ed ogmadsrbtos and distributions, gamma to leads uhsurdnra aits h eodfr aaee wil parameter form second the variates, normal squared such h eodfr parameter form second the oegenerally, More o hnw add we when So, h ewl eett 38a nSc.3372 hr hsmdli model this where 3.3.7.2, Sect. in (3.81a) to revert will We stefu–aaee am itiuin rgnlyment by given originally is distribution, density gamma ga four–parameter the of the extensions several is presented have we 3.2.3 Sect. In is,w ilso o ocmieidnra aitst e a get to variates normal iid combine to how show will we First, parameter form second the of W interpretation the connecting bridge a find can we distribution, distribution: hsrltosi,wihwsepoe yF by explored was which relationship, This n sbt,the both, as and . oictoso h W the of Modifications 3.3 K F ( AUCHON IES x Y 1. 2. ) j ν sicesn for increasing is ersnsaohrwyt obytuct h W the truncate doubly to way another represents Y p f ∼ iid ere ffedm sthe As freedom. of degrees 1 ( Y Y , x No a.3/1 Tab. Y 1 2 | ta.extension al. et 1 ,b ,d c, b, a, 2 + Y , (0 ∼ iid Y f h σ , X 2 2 ( ( 2 χ x x No ν 2 .For ). ∼ iid = ) = ) = ) 2 ∼ 2 = j , = ) n h W the and – (0 σ No 1 β χ σ 2 σ , 1 = qae omlvrae hc r etrdado qa vari equal of and centered are which variates normal squared b X d (0 j ≥ EIBULL 2 =1 ν Γ( 2 = (2) = ) 1 = β λ β λ σ , d c , 1 2 d Y n(.2)awy suiy f nta of instead If, unity. is always (3.82a) in n thsabttbsaefor shape bathtub a has it and , 2 , . . . , ) j ⇒ ( = ) 2 ( ehv h he–aaee W three–parameter the have we b b  We ( and distribution − EIBULL − b x ( ⇒ a χ Y 2 − a a − 2 b 1 0 = k ( ) exp ) 0 2 dsrbto srltdt omldsrbto nthe in distribution normal a to related is –distribution x Y a σ , a + ) j x = +  β b , ⇒ − +1 Y √ ∼ iid d c itiuin,aelne i h eea gamma general the via linked are distributions, ( q 2 c 2 2 a − d ) 2 = ( − , No a ) Y 1 AUCHON b 2 n(3.82a). in β λ 1 ∼ +
2 − exp − (0 c , 1 +  ( t u u v x see σ c σ , . x  b ) Y 2 β X j 1 = EIBULL 2 2 − 2 − − +1 χ =1 2 EIBULL a.3/1 Tab. = ) k 2 x  a ta.(96,gvsa interesting an gives (1976), al. et (2) ∼ Y d , sfrseilvle fteparam- the of values special for es  x j ⇒ 2 β , We xeddb fhparameter. fifth a by extended s − oe yH by ioned m itiuin n fthem of One distribution. mma b ) = ∼ be l 0 n h omldistributions. normal the and X ) a Sc.3.3.5). Sect. (see distribution (  . ν/ < β < he–aaee W three–parameter x EIBULL  We a, c ∼ d − 2  =  ie a gives 2 a ν ; χ a, ) σ β k 2 2 = x 2 − 1 ( : ARTER distribution,
ν ≥ h prahof approach The . 2 1 1 ) eadd we , /c σ , , a. χ 2 c , 2
–distribution 1  /c 16) Its (1967). . ,k c, , EIBULL ν (3.81b) (3.82a) (3.81c) c 2 = ance,  1 = . 165 k Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC nfitn h itiuint ie aaes nta fonl of Instead datasets. given to distribution the fitting in ... ieprmtrdistributions Five-parameter 3.3.7.2 section. following the in itiuinhsdfeetexponents different has distribution v–aaee xeso f(.2)gvnb F by given (3.82a) of extension five–parameter A exp–term: h aadrt eae sflos When follows: as behaves rate hazard The h aadrt a ahu hp when shape bathtub a has rate hazard The zdgmadsrbto n t eaint the to relation its and distribution gamma ized h F The β emninsm oet eogn o(3.82a): to belonging moments some mention We 166 P P F 1 AUCHON HANI HANI F – f h • • β ( ( ( x x x 2 AUCHON ∗ ∗ ∗ is it Otherwise ∗ ∗ (1 cmiain edt nicesn aadrate. hazard increasing an to lead –combinations = ) 1 = ) = ) 18)hsitoue fhprmtrit 38a norder in (3.81a) into parameter fifth a introduced has (1987) ta.extension al. et eraigfor decreasing for increasing for constant ahu–hpdfor bathtub–shaped piedw ahu–hpdfor bathtub–shaped upside–down − c d ta.extension al. et λ λ )  − ( ( ta.(96 xeso ntepeeigscinrssuo t upon rests section preceding the in extension (1976) al. et Var [ E x x c E exp ( − − X ( ( c X X c − a a r 
) ) = ) = ) 1 = > c < c β β − 1)] 1 1 λ − − = , 1 1 1 > ( ( 1 > c x b ,   . ( 1 − ( a  − ( Γ( β b ( β b hni is it then , b b b + Γ( x 1 a r − d − 1 , 1 ) ) b ) b β β d − x − 2 x 1 X ) i Γ β =0 )  ) r  β a 1 β  β a β 2 2 2 , Γ( d und +1   +1 2 2 Γ( + r 0 ( + ) i d 0 ( + )  ) ≤ d < c < β 1 c 0 Γ )  2  a χ β β < β a b o h oiao n eoiao nthe in denominator and nominator the for  2 , 2 ≤ 2  dsrbto,telte en h u of sum the being latter the –distribution, AUCHON − d − r x 1 − + 1 . β β i ≤ 1 < 1 Γ 2 c ) ) < b   x n exponent one y x 1  d  Γ ta.(96 ilb presented be will (1976) al. et . and + ∞ 2 exp  c i , d  0  + , oati oeflexibility more attain to − β < 1 c λ eae distributions Related 3  ( ( 2 x b . β − − < n(.1) this (3.81a), in x a 1 ) ) β β l other All . egeneral- he 2 1  (3.83b) (3.82d) (3.82b) (3.82c) (3.83a) (3.83c) , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC aet eeautdnumerically: evaluated be to have h e,fit,prmtri hsW this in parameter fifth, new, The n orsodn DF corresponding and h a oet fti itiuincnb xrse ythe by expressed be can distribution this of moments raw The ihnncnrlt parameter non–centrality with where with o fw tr with start we if So, factor scale and ihaCFgvnby given CDF a with .. ehv non–central a have we i.e., nqa ozr u r tl oocdsi,w nwthat: know we homoscedastic, still are norm but the zero instead, to If, unequal variates. normal centered and squared W the of Modifications 3.3 Y j ∼ F f ( ( No K x x = ) = ) ( ( · µ ) j steK the is e e σ , − − λ λ 2 ); c b 2 c f k  Y j ( ( UMMER ( α β j y 1 = x X j E ) ) ∞ = = ) =0 X j j − b X , No EIBULL = j a 2 := := η r Γ( ! χ , . . . , 
k fnto EE K (E.E. –function ( a 2 k c µ –distribution: y = + λ j j k − β α K σ , j − + b 1 distribution t u u v 2 ( ( c r 1 ( β k α 2 exp ,β z β, α, k e EIBULL ) X exp j 2 − ) 1) + eget we , 1) + =1 λ n independent and k λ ( =  x  Y Γ Z a η − − j  − . . . := ) 2 . . . 2 a 2 = Γ( k )  1 ∼ σ c ( itiuini h o–etaiyparameter non–centrality the is distribution  + ( λ 2 ( k x β α We σ ) X t j X UMMER j + − ∞ r 2 c =0 b =1 + 2 − + k b  . y η a j a j ,η a, µ ( ( K  α β  − j 2 − ) c  )

) c  − j k j 1) = ( 80–1893): – 1810 , 1) X j k z 1 j X ∞ j ⇒ =0 + + /c , ∞ , j ! =0 j lvrae aedfeigmeans differing have variates al ,k λ k, c, , ) r ( c ( j − Y β α j ,λ k, , Γ( ! 1 ( Γ( ! ) ) y λ η 0 = 0 exp j =1 := K =1; := λ j X j 
+ 2 j =1 UMMER +  k , ) j k − . Y k ) ) j  2  ∼ t x − –function b σ − b a 2  χ a c 2   (2 k c (3.84b) (3.84a) (3.84c) d ,λ k, t . but , 167 ) λ , . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ih(.5)adusing and (3.85a) with parameters The where ie n opudW compound and Mixed W 3.3.8 168 aito sa admadcno eatiue osm other some W the to where attributed approaches some be present cannot will and we section random at is variation tef nSc.3382teW the th 3.3.8.1 3.3.8.2 Sect. Sect. In in variables. other itself; more or one on dependent o–eraigfnto of function non–decreasing a upse oegvnciia value critical given some surpassed Let time. on aibe te hntm,called time, than other variables eeaiain fteW the of generalizations ihCDF with ie by given W two–parameter a is time of point each at which uain efis rsn napoc fZ of approach an present first We duration. h cl parameter scale The parameters Time-dependent 3.3.8.1 Z UO Y ta.spoetefloigW following the suppose al. et ( b t ) ( t ethe be ) EIBULL and c degradation ,β α, ( t ) r pcfidas specified are b itiuin ihvrigparameters varying with distributions R f and n/rtesaeparameter shape the and/or ( 2/1 Tab. ( t t EIBULL EIBULL exp = ) = ) γ R aet esc that such be to have b c exp = ( ( ( fadvc ttime at device a of EIBULL t t t t = ) h eiewl o ala oga h erdto a not has degradation the as long as fail not will device The . = ) = ) earv at arrive we − F covariates itiuin r oeswt ayn aaees u that but parameters, varying with models are distributions itiuinweetelte stemdlfrastochastic a for model the is latter the where distribution ( Pr H d y D EIBULL ( ( exp ( | ( hs h eiblt ucin(CF fadvc is device a of (CCDF) function reliability the Thus, . T t t aaeeswl emd eedn nmeasurable on dependent made be will parameters − − t α α = ) = ) ≥    ( β b b + 1 − D D t . ( (  Pr exp( UO = ) t t H ihrsetto respect with CHR d  ) ) b  t D   ( Y b 1 t c c t D ta.(99 ihboth, with (1999) al. et ( t ( ( Pr H t γ )  ( t . t t t )  ) ) β c Y EIBULL  ) ) ) (  c ) t Y aebe aetm–ayn nsome in time–varying made been have c ( , exp( ≤ ( ) ( t , t ) t ( d ost nnt as infinity to goes ) ) EIBULL t , y )  ) sasmdt earno variable random a be to assumed is  D γ/t ≤ ,  itiuin fcourse, Of distribution. d D b ) xlntr aibei time is variable explanatory e t ( .  t ) . aaeesaefunctionally are parameters  c itntvral.I this In variable. distinct ( t t : ) . eae distributions Related 3 b t and ∞ → c dependent , Starting . (3.85b) (3.85d) Y (3.85e) (3.85c) (3.85a) ( t ) is Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ihtefloighzr rate hazard following the with ewe 38a n 35d sta ehv w subpopulatio two have we τ that is (3.54d) rat and hazard (3.86a) joining between tw continuously a with of model case risk special competing a a be to recognized easily is approach This alr fti oe are model this of failure oiindb W a by positioned pt h hnepoint change the to up alr fti oe are model this of failure (see model sectional or posite . oictoso h W the of Modifications 3.3 Introducing between changes aadrate hazard Z Z S 1 RIVASTAVA H ACKS ACKS F ( f H R t f ( ( ( ) ( ( ( t t t and t = ) t 1 = ) t = ) = ) exp = ) = ) 18)peet W a presents (1984) al hsan this calls τ 2 h ( τ t (        i 1 b 17)suisaW a studies (1974) ) t (  ( ) − t nta fol n variable one only of instead τ e ); b is 1 τ − c b b 1 exp ( b ( 1 t/b EIBULL i t 1 +  h ) − 1 = ( τ and  exponential–W  (  t 1 c b  = ) − ( 1 + 1 b t b t t τ t τ , 1 h ) 1 − b EIBULL 2; b − b + (  ( h eiehsacntn aadrt htatrad ssupe is afterwards that rate hazard constant a has device the 2 1        aadrt.Ti oe sntigbtaseiltofl com two–fold special a but nothing is model This rate. hazard t c t trbtdt w ifrn odtoso sg ftedevi the of usage of conditions different two to attributed , τ c  ) stecmltv ie,satn at starting times, cumulative the as − = ) EIBULL     1 τ c c 1 b et 3.3.6.3 Sect. c − + 2 t b ( 1 2 + b t − distribution EIBULL c t    b 1 b for for ) − − c b 2 τ b  c  c τ > = oe hr h hp aaee sgvnby given is parameter shape the where model    τ  − EIBULL + 1  τ τ b 1 τ, > t 0 ( t 2 1 2 + b 1 b b  , t ( ( − 2 2 for for ) c ≤ b t oe ihasaeprmtrwihperiodically which parameter scale a with model t   )  ) c τ .Sm te fu other Some ). − c   t t − t , c c 1 oeohrfntosdsrbn h ieto time the describing functions other Some .  ≤ − − 1 ) τ > t 0 c distribution 1 1 − + ≥ ≤ ) τ ) 1 exp b 0 ,        t c exp 2 , for for ≤   τ  τ − 2 b − ( 2  τ. > t 0    t )  ntefis hs flifetime of phase first the In . ≤  s e 35d.Tedifference The (3.54d). see es; cin eciigtetm to time the describing nctions –opnn eissse or system series o–component τ swt needn variables independent with ns t c 1 − t b − ( b 1 1 t ≤ t ) τ . 0 =  τ  c + −        pn nphase in spent ,  c   τ 2 t , b ( 2 t ) ≥  c  0 , . (3.86d) (3.87b) (3.86b) (3.87d) (3.87a) (3.87e) (3.86a) (3.86c) (3.87c) i the , 169 ce. r– - Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC time bv ms eadcesn ucinof function decreasing a be —must above function The ate onw ilso htteW the that show will we down Farther nrlaiiyapiain upeetr variable supplementary a applications reliability In h eea omo h W the of form general The of function some is lifetime its that and hre h iet alr seSc.6,tu xliigt explaining so thus of Sect.16), application (see the failure to items time technical the of shorten testing life In etc. hnteW the When covariates with Models 3.3.8.2 htepano nunetesellnt.Tosc approach such Two length. spell the intro influence to or opportune explain often that is it variable, duration other some where 170 with W 31 32 EIBULL • • • cie h eednyo eprtr ntesedo reacti of speed the on temperature of dependency the scribes ugse edn o hsscin L section: this for reading Suggested t BEL ..A S.A. The ie the gives ls sthe is class h aadrate hazard The ceeae iemodel life accelerated h cl parameter scale The h ihrtesrs xrie na tm he ye of types Three item. an on exercised stress the higher the ( y ) rz nceityfrhster feetoyia dissoci electrolytical of theory his for chemistry in prize + µ A rprinlhzr model. hazard proportional RRHENIUS r ′ =max(0 := ( RRHENIUS EIBULL ,τ c, M β ( A j s = ) ( RRHENIUS rprinlhzr model hazard proportional ) c = ) steso–called the is 15 97 a wds hsc–hms h n10 rec — 1903 in — who physico–chemist Swedish a was 1927) – (1859 r itiuin rayohrdsrbto,i sdt oe li model to used is distribution, other any or distribution, y , h ! F            ( –equation ) t ( 1 ) Z . t j EIBULL b − | sdrcl eedn ntecvraes.Tepooyeof prototype The covariate(s). the on dependent directly is Z s 0 siflecdb h upeetr aibes.Ti ie t gives This variable(s). supplementary the by influenced is ACKS –W ∞ . 1 = ) e − x 1 j EIBULL /b E − 31 32 1 ceeae iemdlwt n tesvariable stress one with model life accelerated ie h olwn oml o a moments raw for formula following the gives AWLESS − β X X j s ceeainfactor acceleration exp =0 ( r h tesmyb lcrcl ehncl thermal mechanical, electrical, be may stress The . exp r s
exp( = ) EIBULL b  = j  distribution 1 s 1 − 18) N (1982), j edn oahge rbblt ffiueu to up failure of probability higher a to leading , − b ! r    x µ . + β b r ′ + α ceeae iemdli dnia othe to identical is model life accelerated ( e ,τ/b c, 0 t x r ELSON ( s + s c X j ersnsthe represents )
=0 r α  no oekn fmaterial. of kind some of on . c ) 1   to.(.9)i t rgnlvrinde- version original its in (3.89a) ation. d /s (1990). eterm he ueoeo oeohrvariables other more or one duce x hc i h parametrization the —in which j r ekn fsrs samasto means a is stress of kind me t ,  ) swl ediscussed be will es τ for for ≥ r β − ( j 0 ceeae ietesting life accelerated s . M j j ) stress 0 = ≥ r xesvl used: extensively are j eae distributions Related 3 ( c 1 ) . nteie such item the on          ie h N the eived eieor fetime s (3.87g) (3.89a) (3.87f) is (3.88) this he O - . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC hs h ai ftehzr ae o w tm ihcovaria with items two for rates hazard the of ratio the Thus, hc sidpnetof independent is which and o fa tmwt stress with item an if So, ie by given h oet eogn o(.8 r h aea hs facomm a of those as same the are (3.88) to belonging moments The Because rprinlhzr oe scaatrzdby characterized is model hazard proportional A where omdltesjuntm fa niiuli ie tt,e. state, function given scalar–valued a the in of individual argument an The of job. given time sojourn the t model in to applied extensively been has model hazard proportional h aadrt orsodn o(.8 is (3.88) to corresponding rate hazard The naclrto atrwihdpnso vector a on depends which factor acceleration An . oictoso h W the of Modifications 3.3 W EIBULL • • s 2 The n ed othe to leads and h equation The ie the gives ecnstate can we , h 0 β ( t ( E itiuinwt h cl parameter scale the with distribution ) s YRING ) scle the called is sspoe ob eraigwith decreasing be to supposed is oe W power –equation E YRING t EIBULL s EIBULL β aeiehzr rate hazard baseline okn tteCFo 38)frtodfeetsrs levels stress different two for (3.88) of CDF the at Looking . 1 ( s a lifetime a has h exp = ) F ( H E (H. t –W ( | t distribution s h h | distribution h = ) s EIBULL ( ( β ( t t 1 YRING t ( T ( = ) | | s | 2 s s = ) s β b X i 1 2 = =1 = ) m β ) ) F c ( T ( = s β β α s γ 91–18)i natraiet (3.89a): to alternative an is 1981) – 1901 ,  1 = ) distribution ) g ( ( i hna tmwt stress with item an then , 0  s s β β ( s  s . 1 2 s i β β fe t nrdcinb C by introduction its After . ( ( α b ) ) ) ) s s s β b α ( ( 1 ewl have will we , h elcdby replaced 2 1 T s s s 0 exp = ) ) exp( 1 0 2 1 /s t ( of ) ) ( . t s t 

γ ) ) c m 1 g s ,  α , 2 ( c . ·  2 tesvariables stress − ) esca n cnmcsciences economic and social he γ s 1 . a eavco o.Temost The too. vector a be may ) e values tes ′ . s β b . nuepomn ri a in or unemployment in g.,
. T ( 2 s ) T < . s s 1 ntwo–parameter on 1 2 for s and a lifetime a has i sgvnby given is OX s s 2 2 s > 17)the (1972) is (3.90b) (3.90d) (3.89d) (3.89b) (3.90a) (3.90c) (3.91a) (3.89c) 1 . 171 T s 1 2 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC with ihCDF with When ihtm;ie,srs sgoigwt ie hsapoc i N approach period. This testing time. the with m growing is life stress accelerated i.e., the time; with of generalization a mention finally We om for forms model h W the uttd fmtos see methods, of multitude which for form log–linear the perhaps is one popular of constancy the to ti loisrciet iw(.8 hog h distributi the through (3.88) view to instructive also is It cide. with oainsaergeso model regression location–scale eeainfco nms eea om—is — of form distribution The general variables. most in — factor celeration where of DF the is (3.92a) way, another Written oua rprinlhzr oe ae h W the takes model hazard proportional popular 172 ae unn 39a into (3.91a) turning rate, γ EIBULL 0 Z hr h tesi asda iceetm intervals. time discrete at raised is stress the where ln = a eue xrm au itiuinwt DF with distribution value extreme reduced a has β f ( ( s y b ) | rprinlhzr oe n h W the and model hazard proportional or s and = ) a ∗ b β ( ELSON ∗ s b ( 1 ∗ ) n(.2) so (3.92b), in s F = ) sotnepoe oehrwt ihr(.8 r(.2) Th (3.92a). or (3.88) either with together employed often is exp ( t L | b AWLES 19)soshwt siaeteprmtr o a for parameters the estimate to how shows (1990) s  γ ∗ 1 = ) Y ′ y = s h so xrm au om(e 34ad)adhsDF has and (3.43a–d)) (see form value extreme of is − ( = 1 c iherror with t b a Y − ∗ | a i ∗ and P s g ln =1 m ∗ 18,p.299ff.). pp. (1982, ( exp = ) ( = ( s 1 s s T ) γ ) = ) a i a −  a osatvrac.Avreyo functional of variety A variance. constant a has ∗ = s g ∗ ( i ( − ( exp s hslna ersincnb siae ya by estimated be can regression linear This . s s γ  Z + ) g ) β ln = ) 0 ( h osac of constancy The . b c ( + s  β EIBULL s ) y ( b  ) γ  s ∗  − b c t ) ′ Z, b t ,  .. tdpnso etro stress of vector a on depends it i.e., , s  EIBULL b β b a  ∗ c ∗ c − (  aadrt sisbsln hazard baseline its as rate hazard 1 ( s ple nlf etn oshorten to testing life in applied s s nof on ) t , )   dlweetesrs changes stress the where odel . ceeae iemdlcoin- model life accelerated exp ≥ y , Y 0 { . c z ∈ ln = n(.8 corresponds (3.88) in eae distributions Related 3 − R , e z T } hnteac- the when , 39b sa is (3.92b) . step–stress (3.91b) (3.92b) (3.92a) (3.91c) (3.92c) most e Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC esat(et ...)b rsnig—i eal—svrlb several — detail in — presenting by 3.3.9.1) (Sect. start We ewl rtpeetsm iait W bivariate some present first will We eun E a ohmria itiuin sexponential as distributions marginal both has BED genuine andfo iait xoeta distribution, exponential bivariate a from tained multi some on 3.3.9.2) (Sect. overview an giving by finish and n M and ... iait W Bivariate 3.3.9.1 ntal h w opnnshv osatfiuerates, failure constant have components two the Initially pnacerydfie hsclmdlta sbt ipeadr and simple W both possible is study that to model physical defined syst clearly two–component a in upon life component’s of dependence tical hnbt r noperation: in are both when Wsbsdo F on based BWDs u h lifetimes the But esoso h n–iesoa W approache one–dimensional two the mainly of are tensions th There distribution multivariate distributions. a marginal f its need the we systems When dependent, series are components. nents the failing and independently parallel of o assumption the distribution lifetime i.e., the system, studied have component we 3.3.6.2 Sect. In o euti elcmn,btcagsteprmtro the of parameter the changes but to component replacement, a in result not .. utdmninlW Multidimensional 3.3.9 W the of Modifications 3.3 J F OHNSON REUND 33 • • 19) M (1990), ugse edn o hsscin C section: this for reading Suggested S B a W has eydfeetapoc st pcf h eednebetwe dependence the W specify ate to is approach different very A h rnfraino oemliait xrm au dist value extreme multivariate some of transformation the o nqe hr xs eea xesosalhvn margi having expon all multivariate extensions or several bi- exist multiv there ext a and the unique, to bivariate As not distribution the exponential transformation. transform power ate to through is distribution tial approach simple One PURRIER K ALAKRISHNAN ARSHALL OTZ 16)pooe h olwn alr ehns fatwo–co a of mechanism failure following the proposed (1961) 20)so,mn fsc Eseit h E oeso F of models BED The exist. BEDs such of many show, (2000) EIBULL /B EIBULL ARSHALL /W λ ALKRISHN i ∗ f EIER /O otyto mostly , i X ( REUND /J x LKIN EIBULL 1 aitss htteeegn i rmliait W multivariate or bi- emerging the that so variates i OHNSON marginals. 18)adT and (1981) /O | and EIBULL λ LKIN i EIBULL = ) 16)hv eevdtems teto ndsrbn h st the describing in attention most the received have (1967) X sBED ’s xesost hs w BEDs. two these to extensions λ 2 16) P (1967), 20) L (2000), λ i ∗ /J r eedn eas alr fete opnn does component either of failure a because dependent are i distributions ARAMUTO λ > OHNS EIBULL exp distribution EIBULL ROWDER i {− ATRA EE stennfie opnn a ihrworkload. higher a has component non–failed the as EIBULL λ 17) L (1979), /W i /D models 20,Ca.4) iia prahinvolves approach similar A 47). Chap. (2000, x ADA EY distribution: i 18) H (1989), } ; 19) R (1999), BED (2001). distributions, x U i 33 18,19,19ab,L 1992a,b), 1990, (1989, ≥ ANAGAL o hr,b oe rnfrain A transformation. power by short, for 0 OY λ , λ /M i 1 w pca ye famulti– a of types special two f ti o ipytepoutof product the simply not is at > oitouemliait ex- multivariate introduce to s UKHERJEE and sK As . 19) H (1996), ne h ahrrestrictive rather the under , m.Teetossesrest systems two These ems. BWD ait W variate iedsrbto fteother the of distribution life 0; aitc o ti appropriate is it So, ealistic. iuetmso h compo- the of times ailure vraeW ivariate λ ribution. i 2 1 = ihidpnetDFs independent with , ntoo oeunivari- more or two en OTZ nino h univari- the of ension a xoetas see exponentials; nal o hr,ta r ob- are that short, for OUGAARD nildsrbto is distribution ential 18) S (1988), EIBULL , /B 2 EIBULL U . pnn system: mponent ALAKRISHNAN /B raeexponen- ariate EIBULL REUND HATTACHARYYA ARKAR 18) K (1986), distribution models. models (3.93a) (1961) (1987), atis- OTZ 173 / / Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC Provided 39e sntepnnilbtrte itr ftoexpone two of mixture a rather but exponential not is (3.93e) enwsbttt h xoeta itiuin n(3.93a) in distributions exponential not the do substitute now (3.93c,d) We So average. weighted a is (3.93e) otherwise olwn parametrization: following where h itiuino h iefo rtfiuet alr ft of failure to parameters λ with failure distributions first exponential two from of time mixture the of distribution The hr sn te eedne h iet rtfiuei expo is failure first to time The dependence. parameter other no is There 174 h rbblt htcomponent that probability The ucinCD is CCDF function hnoecmoetfis h eann ieieo h othe the to of equal lifetime parameter shape remaining The component. the fails, component one When f f W f R f 1 1 1 R F R F i , , i  EIBULL ( 1 r F 2 2 ( x , λ 2 y ( ( i ( i epciey ial,tejitD of DF joint the Finally, respectively. , x x = ) x | 1 1 1 λ x , x , x , λ i λ c , u ihpsil hne parameters changed possibly with but , λ i 2 2 1 2 ) = ) 1 eeal,wihmkstehzr aeo h o–aldcom non–failed the of rate hazard the makes which generally, i )= + = ) = sntcniuu at continuous not is + λ                            λ 1 2 c 2 λ λ λ f λ f i 6= − 2 1 1 1 1 2 λ ( ( + + λ λ i λ x x λ 2 ∗ 1 ∗ i ∗ 2 1 i ∗ λ λ λ h agnlD of DF marginal the , 1 1 | | exp exp i    2 2 λ λ y − − 2 1 i ( ) )
  λ λ λ c R R λ i i − − 2 ∗ 2 ∗ R − i ∗ − 1 2 i ( ( 1 ( ( h h λ i λ λ λ λ stefis ofi is fail to first the is x x ( λ λ λ 3 x c exp 2 ∗ 1 ∗ x 2 1 2 1 − 2 1 i ∗ i ∗ x x i − − | | ( ) i = exp exp ih eeulto equal be might | 2 1 λ λ exp  λ λ λ λ λ 2 − 2 − x i − ) ) 2 ∗ 1 ∗ 1 exp = ) 1 =   exp ) exp ) f f ( ( +  ( λ λ 1 2 = − − λ λ ( ( X 1 1 − λ i x x X 1 ( ( + x + y 2 1 1 2 λ λ λ +   i exp ) 2 i and ) 1 1 − − i ∗ λ λ unless , is  c − − λ x + + 2 2 i λ x x

2 − i − − ( (

i ∗ . 2 1 X ; λ λ λ λ λ  λ | | and 2 2 i λ λ 1 1 2 y  λ λ i − − − 2 ∗ 1 ∗ c i + + is x λ 1 ∗ 2 ∗ ) ) i λ ≥ ) ) u h cl parameter scale the but , ( i hnvrtefis alr occurs. failure first the whenever , λ c x x λ λ 1 λ λ

λ yW by i ∗ 1 ∗ 1 2 1 ∗ 2 ∗ λ 2 2 . 0; 1 iei ee ntesurviving the on reset is Time . for for

) ) ta itiuin for distributions ntial ) ) or 2 ∗ eohrcmoeti hsa thus is component other he x + ersn eun BED. genuine a represent x x x λ 2 = 1 for for λ 2 1 λ EIBULL i −

0 0 − c , 2 ∗ etal itiue with distributed nentially 2 λ i i ) nproportions in ≤ ≤ λ i λ 2 0 0 x 1 ∗ > for for 2 ∗ λ i opnn sstill is component r x x x ≤ ≤ eae distributions Related 3

x 1 ∗ 2 1 1 0; h on survival joint The . 2

+ x x itiuin nthe in distributions x < x < 0 0

1 2 i + ≤ ≤ +    x < x < 1 = oetchange. ponent 1 2 x x    ; 1 2 , 2 1 x x < x < 2 ,    λ i . λ λ 2 ≥ (3.93d) (3.93b) i i ∗ (3.93e) (3.94a) (3.93c)  2 1 λ > λ snot is . 0                      . and i ∗ , Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC a ob eemndb ueia nerto f(.6)ove (3.96c) of integration numerical by determined be to has h on Fof DF joint The h Fo h iet ytmfailure system to time the of DF The 39b un notefloigBWD: following the into turns (3.93b) stm srsto h uvvn opnn,i a hfe W shifted a has it component, surviving the on reset is time As W the of Modifications 3.3 fF of 39b ils(.3)frteseilcs falsaepara shape all of case special the for (3.93c) yields (3.94b) iait W bivariate nte iait W bivariate another rmtefis otescn alr.S h ieo ytmfail system of time the So failure. second the to first the from U hp parameter shape ,θ> θ λ, θ (see S L f PURRIER U 1 34 ni h rtfiuei itiue stemnmmo w ii two of minimum the as distributed is failure first the until , 2 18)sosta h bv W above the that shows (1989) REUND f ( et 3.1.4 Sect. h hf rlcto aaee seult h ieo h firs the of time the to equal is parameter location or shift The y ( 1 ,w u, f y , W 0 hnw aetefloigcniinldniyof density conditional following the have we Then . 2 | /W u = ) )= EIBULL sBDrsligin resulting BED ’s ( f w 1 EIER ∗                    , .Uo h rtfiuetermiigcompo remaining the failure first the Upon ). = ) f 2 U ( U f c f c c y 2 ∗ 1 ∗ 1 2 ( n cl factor scale and and 1 ( ( u λ λ θ λ c 18)mdfidteF the modified (1981) y , oe bandb sn ietpwrtasomto fthe of transformation power direct a using by obtained model y y f EIBULL ) 2 ∗ 1 ∗ 1 2 U   f 2 | | λ λ ( W = ) W λ λ u  2 ∗ 1 ∗ 1 2 2 = ) EIBULL λ | ( ( is c , c , u y y (                  ( 2 1 w θ 1 2 oe.Let model. w − − exp ) exp ) 2 = ) f f λ c y y 1 2 f f + distribution 1 2 ( ( 2 1 ) ) y y ( ( ( u λ     1 2 y y EIBULL u λ exp V c c ) so , 2 1 2 ∗ | | 1 ∗  − − − − λ λ exp c | | = ) − λ λ {− 1 1 1 2 c c 1 c , c , − 2 ∗ 1 ∗ U 2 exp exp λ λ U c , c ,  1 2 1 λ θ exp REUND ( 1 2 + − etetm otefis alr and failure first the to time the be u λ exp ) exp ) y y 2 1 xeso oF to extension exp   1 2 ) )  W 2

 − − λ c ) c c − c  for for u 2 1 ( osnti eea aeacoe omand form closed a have general in not does }  

w θ   c  λ λ − − λ dao aadrt hnet derive to change rate hazard a of idea − − × 2 ∗ × 1 ∗ 1 0 0 ( 2 ( ( (   + ( w θ y y ( ( ≤ ≤ w θ λ λ alr facomponent. a of failure t 2 1 u λ u 2 1 − − y y ) + 1 2 − − c c  eeseult one. to equal meters ) + REUND 1 2 y y c c h set the r etcagsissaefco to factor scale its changes nent u

1 2 λ λ y < y < − r is ure u ) ) ) u , 2 ∗ 1 ∗ W )    ) ) ( c c c EIBULL W d u λ 2 ∗ 1 ∗ y y 2 1

× c c |

2 1 1 2 u sBDi o qa oa to equal not is BED ’s w , ≥ .   V : ) c c { for for c 1 2 EIBULL 0 (

=

,w u, ≥ ; × × U 0 0 distribution ,w u, 0 ≤ ≤ : ) + .                  y y W aitswith variates u 1 2 W ≥ + y < y < h time The . marginals 0 w h time the . (3.96b) (3.94b) (3.96c) (3.96a) 2 1 34 (3.95) = .                    and 175 v } . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC f : have we II and I cases For h hsclmdlbhn h M the behind model physical The L that mention finally We nti aew get we case this In hcsi oendb he needn P alway independent is three which by shock governed a follows: is receiving as shocks is after model fail their system precise, component more be To multaneously. and aeIII: Case Wsbsdo M on based BWDs failure. first compone the upon two parameter shape the the for of change parameters shape different introducing h agnldsrbtosaegnieoedmninlex one–dimensional genuine are distributions marginal The N N aeII: Case variates. R is: components, two the of lifetimes λ aeI: Case are (3.96c) of cases special important Three 176 UND iigtema ubro vnsprui ftm.Eet nt in Events time. of unit per events of number mean the giving , ( 1 MO 3 ( v , ( t 2 = ) t ; N sBDisfra hr satidfiuemcaimta hits that mechanism failure third a is there as insofar BED ’s ; λ ( λ 2 x = ) 3 ( 1    t ) x , ; c r hcst ohcmoet.Tejitsria ucino function survival joint The components. both to shocks are c λ { θ 2 4 2 2 ;= 1; = 1; = N = ) ) λ θ λ ;= 1; = f 2 ( r hcst opnns1ad2 epciey n vnsin events and respectively, 2, and 1 components to shocks are = exp = = t ( v (2 ) v t , exp 2 = )    Pr Pr θ ⇒ − ARSHALL ⇒ ≥ (  = 1 = X exp exp θ  N  h oe eue oarprmtiaino h F the of reparametrization a to reduces model The ) 0; R 1 − − λ c h ieie ftetocmoet r i W iid are components two the of lifetimes The 1 − ( i x > 1   ( x λ 2 λ U c x exp 1 − − } v λ 1 v i ⇒ ; ema P a mean we , 18)hsgnrlzdtemdlo S of model the generalized has (1989) exp = ) 1 c x λ λ ( /O − X , λ 1

1  1 1 0 = ) 1 ; h ieie ftetocmoet r i exponential iid are components two the of lifetimes The − x  − LKIN ARSHALL 2 + 1 exp v λ x > v λ θ − λ 2 ≥  N , 3 x {− ( sBED ’s ) − λ 2 2 0  x ) 2 − and 2 ( ( 1 v λ + ( λ OISSON /O 1 − x λ OISSON i 2 λ 3 − + ) θ λ ; LKIN 3 c max[ λ 2 } 2 = ) exp λ 2 x x exp 3 0 = ) 2 2 )  .

rcs (see process 16)BDdfesfo hto F of that from differs BED (1967) x x processes {− i − 1 for for

x , N , ; (2 ( 2 2 i 0 0 ] − v λ 3 oeta distributions: ponential

1 = (max[ ≤ ≤ . v λ θ ) t n yas loiga allowing also by and nts h opnnso two– a of components The x x c N aa.Teocrec of occurrence The fatal. s , } 1 2 2 4.2 Sect. i  . ( ) ≤ ≤ x v , t eprocesses he  ; 1 x x λ x , eae distributions Related 3 PURRIER ohcmoet si- components both ; 2 1 i EIBULL ≥ ); . 2 v f ]; REUND    ihparameter with ) 0 i X λ ≥ . 3 1 = 1 0 = ) 0 and /W h process the and variates. , N model. 2 EIER

X 1 , (3.97c) θ (3.97b) (3.97a) ( 3 2 t By . 2 6= ; the , RE λ by 1 , ) - Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC rtW first A h on Fo h M the of DF joint The evs pligtepwrlwtransformations power–law the applying selves, 39ab into (3.97a,b) rte samxueo nasltl otnossria fu survival continuous absolutely an of mixture a as written h atrpoaiiyi epnil o the for responsible is probability latter The x n ehv oiiepoaiiyta ohcmoet fail components both that probability positive a have we and nteform the in n iglrsria function survival singular a and h rbblt htafiueo component on failure a that probability The W the of Modifications 3.3 R 1 35 c R = ( f 1 x 1 MO h orlto ofcetbetween coefficient correlation The , , 1 R 2 x 2 x , ( 2 1 MO y M . ( , 1 2 EIBULL 2 x y , = ) 1 ( x , ARSHALL x 2 exp = ) 1 2          x , = ) = 2 xeso oti oe a ensgetdb M by suggested been has model this to extension λ λ = ) R Pr 1 1          s    λ λ + + ( ( /O ARSHALL X 1 3 x λ λ λ λ λ λ 1 + exp exp 1 i 3 2 1  LKIN x , 2 2 EIBULL λ X < + R ( ( λ − + λ λ 1 exp 2 1 2 MO   λ 1 2 exp = ) + Pr , λ λ 2 2 − − + + 16)so httejitsria ucin(.7,)cnb can (3.97a,b) function survival joint the that show (1967) j 1 3 ( λ  + = ) ( X /O y λ ( ,x x, λ λ 2 distribution exp − 1 c λ 1 1 λ 3 3 1 X LKIN 1 ) ) 3 y ( = − 1 λ R R + ) λ 1   c R and 1 1 1 λ X 1 1 MO MO − − λ c − + , , + 2 2 2 ( 2 3 oe is model x X ( λ y = ) ) ( λ λ ( ( λ λ 1 2 c λ 1 2 y x x 2 i 2 x , 1 2 iglrt ftedistribution the of singularity i 2 1 x is c 1 1 + − 1 + + cusfis is first occurs x , x , 1 + 2 λ ̺ − + ) ( − λ 1 λ λ λ X 2 2 λ 3 3 + 2 λ ) ) 3 3 λ 1 X max[ ) ; 2 X , ) + max 2 λ λ λ y i y for for for x 1 ,j i, 2 3 λ 2 2 2 c c 2 2 = ) 2 = + + 3

 − max[ ) y 1 = λ λ λ x 0 0 0 Y 1 c Pr λ 2 3 3 1 1 nction for for i ≤ ≤ ≤ 3 ( c y , x , + . , simultaneously: X i ; max[ 2; 2 c x x x 1 2 λ x 2 ] 1 2 1 = 3 

0 0 1 i i = x < x < x , ARSHALL R X 6= ≤ ≤ 1 = x s 2 2 x 1 ( ) j y y ] x , 2 1 2 x

; 1 2 c c > 1 2 1 = , 2 x , 0 2; ≤ ≤ ] .

x. 2 y y /O hschanging thus ) − ln h line the along 35 2 1 c c .          2 1 LKIN .             (3.97g) (3.98b) (3.97d) (3.98a) (3.97e) (3.97f) them- 177 e Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 f © 2009 byTaylor& FrancisGroup, LLC h ra nthe in areas The 178 Set by hs r eun nvraeW univariate genuine are These sasltl otnos(e i.31) h on Fi not is DF joint the 3/16), Fig. (see continuous absolutely is iue3/16: Figure b is discontinuity The grid. surface the on discontinuity of h hr rnho 39d ersnstedsrt ato t of part discrete the represents curve (3.98d) of branch third The c 1 1 , 2 = y ( y 2 y 1 c 1 ≥ y 2 or y , 2 h icniut sceryse in seen clearly is discontinuity The . y 2 = y 1 c )= 2 1 /c y to on uvvlfnto faBDo M of BWD a of function survival Joint sraepo n otu plot) contour and plot (surface            1 c 2 1 0 λ λ λ and /c y 1 2 3 n(.8,)t e h agnlsria functions: survival marginal the get to (3.98a,b) in 2 1 − ( ( c λ λ 1 nthe in y y y 2 1 2 2 1 c pae hr h w rnhso 39b r ai,myb ex be may valid, are (3.98b) of branches two the where –plane, + + 1 ≤ − R λ λ 1 3 3 i y y ( exp ) ) 1 c 1 y c c 1 − 1 1 i /c exp = ) y y y  2 1 1 c c 2 . 1 1 − pae hslcsi tagtln hog h rgnfor origin the through line straight a is locus This –plane. EIBULL − − 1 1 ( c c λ 2 2 1  y y + 2 2 c c − 2 2 λ − − itiuin.Weestejitsria function survival joint the Whereas distributions. ( 2 1 1 λ + exp exp i i.3/17 Fig. λ + 3   ) λ y − − 3 1 c ) 1 λ ( } λ y 1 ARSHALL 0 ; i c 1 y hr ehav we where , i + 1 c

1 i , λ − 3 s eni h otu plot. contour the in seen est ( ti ie by given is It . ) λ y 1 = ejitD n tle nthe on lies it and DF joint he 2 1 c 1 /O + − , λ LKIN 2 λ 3 . 2 ) y y akdsm points some marked e 2 2 eae distributions Related 3 c c 2 2 type

} 0 ; 0 ; ≤ ≤ ≤ y y y 1 c 1 2 c c 1 1 2 = 3 (   y si h rgnlM original the in As . 1 λ Y , 1 − − 1 ; 1 λ y Y , 2 λ λ OISSON 1 1 c 1 1 ftetocmoet are components two the of 1 ; 2 y y c − y > 1 1 1 c c 1 1 0 = ) λ EIBULL − − EIBULL 2 2 processes y ) λ λ 2 c 2 2 N , 2 y y − aadrt.Isrlto oaW a to relation Its rate. hazard 2 2 c c 2 2 2 λ xeso oM to extension ( − − y 3 y 1 2 λ λ c ARSHALL N ; = 3 max[ 3 3 λ . i y y 2 ( y t c , 2 1 c c 2 ; 3 3 . λ 2 y 0 = ) i 1 c , y , for for /O ooeeu P homogeneous e i 2 ) ] ARSHALL ihpwrlwintensities power-law with
N , LKIN c 0 0 3

3 ≤ ≤ sy h on survival joint The usly. . oe’ h processes the model’s max[ y y mtoeo h first the of those om eae distributions Related 3 1 2 EIBULL et nteprocess the in vents /O ≤ ≤ y LKIN y y 1 2 1 y , . rcs ilbe will process OISSON    2 sBDby BED ’s ]; λ 3 (3.99b) (3.99a) c , pro- 3
36 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC W’ ae nL on based BWD’s 39ab aetemarginals the have (3.99a,b) W the of Modifications 3.3 hc,i eea,aentW not are general, in which, emnintefloigseilcsso (3.99a,b): of cases special following the mention We well. as W L U EIBULL /B • • • HATTACHARYYA 18)hstesria function survival the has (1980) tblnst eissse ihtosaitclydepende statistically two with system W series and exponential a to belongs It c function ierhzr aedistribution: rate hazard linear hsi pca case special a is This ytmfiueocr ihcntn n/ricesn inte increasing and/or constant with occurs failure system xeso,weetemriasadteminimum the and marginals the where extension, hc a agnl ftefr ftelna aadrt dist rate hazard linear the of form the of marginals has which c W all are system, series the of and When h nvraemnmmtp itiuin nrdcdb F by introduced distribution, minimum–type univariate The h nvraelna aadrt itiuinwith distribution rate minimum hazard linear the univariate minimum–type. of The univariate extension the of bivariate marginals a has it as because regarded be can which 1 2 R R R 1 = ( = ( ( y y t c oes rtfml fmdl et pntegnrlform general the upon rests models of family first A models. exp = ) 1 2 3 c exp = ) exp = ) c 2 = and 1 2 = R c = 39a un into turns (3.99a) , 1 c , R 2 3 c ( and 2 U ( R y 2 = y /B 1 = R ( 1 y , y y ,    c HATTACHARYYA i 19)hv rpsdsvrlnwcntutoso bivariat of constructions new several proposed have (1990) ewudgttesm iait model. bivariate same the get would we , 1 c EIBULL ( 2 3 EIBULL y , − − − 3 y 2 exp = ) exp = ) 1 = i = exp = ) 2 ( ( ( ( EIBULL exp = ) c λ λ λ 1 2 1 c 2) = R R 39ab oniewt h rgnlM original the with coincide (3.99a,b) , n(.9) egttefloigbvraeetnino the of extension bivariate following the get we (3.99a), in + + + distribution slftm itiuin.Ti oe saporaewhere appropriate is model This distributions. lifetime as ( (  x x λ λ λ  exp = ) exp = )  − 3 3 2 EIBULL fcus,ti xeso a oasltl on DF joint absolutely no has extension this course, Of . fteuiait iiu–yedsrbto.Taking distribution. minimum–type univariate the of  − ) ) + −  y y − ( 1 2 c c λ λ λ y b

λ 3 1 1 3 1 , , (1990) ) 1  y y   t y aitswith variates i c c 1 c 1 3 1 2 − −

− − − − . x a λ λ  λ λ ∗ 2 i 2 x y y b − y 2 y 2 2 − i c 2 2  i − 0

c − . λ T 5 2 ; h λ ∗∗ x b − λ min[ = ( 3 i x 3 x 1 = ψ δ = ) 2 c max[ max[

. . ( st.With nsity. , y 2; RIEDMAN a Y 1 y y tcmoet having components nt y , 1 + 1 1 Y , iuin With ribution. y , y , 2 x b ) 2 2 2  ARSHALL tp distribution –type ] ] ] .. h lifetime the i.e., ;

. a h survival the has c ,

/G c , 1 ERTSBAKH = /O (3.100) c (3.99c) 2 LKIN c 1 = 1 181 e = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC with h iiu of minimum The l W oespeetdu ohr aeasnua component singular a L have by given here is to continuous up lutely presented models BWD All W.Temriaso 311)aealW in all are (3.101b) (3.101a) of of plot marginals contour The BWD. the and plot surface The fti family: this of function the of forms Different 182 • • • • λ so ye(.0)for (3.100) type of is W ob on nL in found be to BWD A δ h M the δ L 1 λ , U 0 = = /B 2 λ > HATTACHARYYA 3 n any and ARSHALL R and 0 ψ ψ f 1 , 1 ( ( , 2 , > c R y y Y 2 ( 1 1 ψ ( 1 y 1 R R y , y , y , 1 ( 2 and 1 ψ y y , 2 1 0 ( 2 2 y , ( ( 1 y ( /O = ) = ) and y y y , 2 y 1 2 2 1 exp = ) Y 1 y , = ) LKIN exp = ) exp = ) 2 y , 2 max = ) 2 0 c sas W also is 2 exp = ) 1 ) γ < rpsdtetofloigfunctions: following two the proposed =  ( EE edt W ihidpnetvariables independent with BWD a to lead EE xeso (3.98a). extension ψ 1  γ exp  c ( − 2 ≤ 2 (1979): y y (1979): b − 1 1 c 1 = exp   y , 2 1 {− {−  λ EIBULL y h orsodn Fis DF corresponding The . − ( − c c 1 2 1 c λ 1 .  1 ) λ λ d /m 1 ( y , λ 1 c 2 γ 1 γ − oehrwt h factor the with together λ λ 1 EIBULL y y y 2 1 c 2 +  y 2 1 c 2 1 γ c γ c ) y
1 c γ ihsaeparameter shape with − 1 y c b  } } + oehrwith together 1 ( 1 + y y λ b . ,  λ 2 1 2 2 λ c : 2  y 1 d 1   2 y c 2 c i.3/18 Fig. y 2 ) 2 c y /m c 2 c
− 2 c ) γ γ 1 − )

1 

− . m λ ; y 3 1 λ , exp y 1 y max hwtecniut fthis of continuity the show 1 2 y , =
 δ c 2 − W en abso- being BWD A . b  il pca members special yield γ c  ≥ 1 − γ d eae distributions Related 3  −      c 1 c . 1 0; 1 y b y λ , 2 2 × 1 c Y d ,  2 1 c 2 2 c = and  y 2 c b  2 − . Y (3.101b) (3.101a) (3.101c) c 2 2 . gives Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC iue3/18: Figure W the of Modifications 3.3 sraepo n otu plot) contour and plot (surface on est ucino h boueycniuu W giv BWD continuous absolutely the of function density Joint EIBULL distribution nb L by en EE 183 Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ... utvraeW Multivariate 3.3.9.2 184 uin(V)fravector a for (MVE) bution hsMDi o boueycniuu u a W has but continuous absolutely not is MWD This nrdcn h oe transformations power the Introducing probability. k ooti oecmatntto o 312) let (3.102a), for notation compact more a obtain To The dimension ( n ftehyperplanes the of one eget we vector h zero–one–variable The ae iutnosyhtb P a by hit simultaneously (are) M W s 1 = 1 ARSHALL EIBULL s , R k R 2 , dmninlmarginals –dimensional ( s , . . . , s ( 2 x x p , . . . , 1 ∈ = ) x , itiuin—MDfrsot—wt on uvvlfunction survival joint with — short for MWD — distribution k S, R /O 2 exp = p ≥ x , ( ) max( y LKIN H . 3 exp = ) 2 hr each where Pr exp = ) i ( j y − ) i − i ijk P =1 R EIBULL in = p exp = ) stemxmmo the of maximum the is − x λ λ ( 1 19)dsusstefloigseilcs f(3.103a): of case special following the discusses (1996) s λ max[ x X 110 x  i 111 s X , of =1 p i j j exp = ) niae hc opnn()o the of component(s) which indicates ( OISSON − x k ( p 2 max[ 0 = λ i i max[ λ aitsgvn h on uvvlfunction survival joint the giving variates x 1 = x > − i ( 6= 100 distributions y i x , − i } x i 101 x x = 1 ; 2 . max( 1) y − s , i i x , sfrwhich for ’s i 1 − x s p y , 2 max[ EIBULL λ r l xoeta.BtteMEof MVE the But exponential. all are 1 = i k ) j s , . . . , λ 12 y ] 2 ( 001 s i c y , . . . , ,j k j, i, − ...p i i p , . . . ,
x x , x S i 1 max[ ) 3 p x , y ) agnl faldimensions all of marginals ) distinct eoetesto vectors of set the denote p λ 2 ≥ . s ] (0 6= ie rtmultivariate first a gives ; ] s
i x − o xml,for example, For . c o 1 = 2 ) p . x , cmoetsse is system –component λ eae distributions Related 3 . , 011 ) 2 Thus, . t. a positive a has etc., , 0 x , . . . , , . . . , max[ 0) n.A least At ent. p x ] o any For . ) 2 ldistri- al x , (3.103b) (3.102b) (3.103a) (3.102a) . 3 p                    ] 3 = Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC hr the where n yfrhrtkn nepnnilydsrbtdvariate distributed exponentially an taking further by and where itr fW of mixture parameter xoeta ihui mean. unit with exponential ν C of MWD the Thus, κ For W the of Modifications 3.3 h itiuinof distribution The W hc sasltl otnosadwihhsW has which and continuous absolutely is which MWD A in a enpooe yH by proposed been has sions h on uvvlfnto is function survival joint The T C P W ATRA ROWDER 1 = 0 = EIBULL min = • • λ o general For . 0 eitoue nte parameter another introduced He h exponent The ( ie 314.Temria itiuin for distributions marginal The (3.104). gives /D a i 0 = 1 EY Y  Y ν c a , . . . , itiuin.Teoedmninlmriaso (3.103b) of marginals one–dimensional The distributions. i 18)hsetne 314 ntoaspects: two in (3.104) extended has (1989) i λ a = and 19)hv osrce ls fMDi hc ahcomponen each which in MWD of class a constructed have (1999) EIBULL 1 1 when R h on itiuini h rdc of product the is distribution joint the X Y j m =1 ( 1 Z im y , . . . , T exp = ) a κ r l needn,te osdrtemliait distri multivariate the consider they independent, all are ) a c ij i ROWDER savco fmxn probabilities mixing of vector a is min( = and itiuin.Seicly ytaking by Specifically, distributions. smd pcfi o variate for specific made is ≥ Y λ a ij R p EIBULL p 0 ν ( Y with y n uhthat such and (" h agnldsrbtosaesc that such are distributions marginal the p X exp = ) R R Y OUGAARD  i ( 1 ( − a h on uvvldistribution survival joint the has y f Y , . . . , t min( = a n–iesoa W one–dimensional a has exp = ) distribution Y i X ( i exp = ) =1 ij z p = ) ∼ ( λ i κ n We > κ Y λ y ) ν 18)adltro yR by on later and (1986) (  .. ftesse’ ieie is lifetime, system’s the of i.e., , i c i 0  − Z , # (0 i − P exp ν =2 − p 0 ) " λ , );

. κ ( a ;  X λ ij i i c + =0 i p  i − c , λ Y 1 = + 1 X i i i /c λ ij =1 λ . p Y > λ i ); 0 ! i 1 = p. , . . . , 0 λ z 0; r ahW each are ) (

t i i y a c p y Z , . ) i ,ν> ν c, c 1 = ij EIBULL i EIBULL c

i needn one–dimensional independent ihDF with # . . ≥ ν , ) 2 0 p, , . . . , ∀ 0; . OY EIBULL ,j i, agnl faldimen- all of marginals itiuinwt shape with distribution y /M κ i are and ≥ + UKHERJEE 0 λ . i P when uinof bution Y i j m c =1 i
ν κ (3.106b) a (3.103d) (3.106a) (3.103c) a a has t − (3.105) (1988). (3.104) ij 0 = κ 1) = ν 185 or is Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC h DF the nti atscino hpe ewl itsm eaie t relatives some list will we 3 Chapter of section last this In Miscellaneous 3.3.10 detail. some in hc ontfi nooeo h lse rmdl etoe abo V mentioned by models proposed or model classes the The of one into fit not do which where 317)gvsagmadniy—se(.4 with — (3.24) see — density gamma a gives (3.107a) n eas fisdpnec nteicmlt am funct gamma incomplete the on dependence its of because — and ntegmafnto in function gamma the on oet r ie by given are moments ehave We 317)i aiyrcgie obe to recognized easily is (3.107a) where in h D eogn o(.0a sgvnby given is (3.107a) to belonging CDF The tion. h on uvvlfnto of function survival joint The 186 x µ 0 ∗ stema and mean the is max( = g ( y | ,c b, x R = ) Var 1 ( x , . . . , x E = ) ( ( Y Y b Γ = ) = ) f = 2.91 Sect. ODA ( p + 1 y G ) c ( | Y Y i i > ( g f =1 =1 X E 0 p p y ( ( 18)—called — (1989) ,c b, , 1 c 1 y 0 x b | b Y Pr X
j X , . . . , h pca case special The . 2 ,c b, 1 | m =1 | /c /c r ( 1 + Γ(1 ,c b, (  b, a ob vlae yn by evaluated be to has — )
) X = ) y a c ( 2 + Γ(1  )= 2) steD 28 facnetoa W conventional a of (2.8) DF the is = i = ) ij  ( 1 + Γ(1 ( 3 + Γ(1 x > c = b p exp exp r/c ) µ /c b y is x i Γ Z ∗ 2 ) Γ  Γ /c ) n  0 √ f − Pr exp( + 1 y Γ − b /c /c ) ( + 1 c g y (  pseudo–W y ,  + 1 ( x > Z ) ) | y λ y c 1 c − 0 − m + 1 ij r |
,c b, ,  +1 b x/b ,c b, c c 1 x c Γ Γ = o and
i c
2 2 0 1 ij c ) ; ) 1+1 + (1 1+2 + (1 ) p )
. , . d  + 2 = d ,c > y c, b, u 1 EIBULL /p h W the o 2 = x λ r ion ve. /c /c ssuidb P by studied is mrclitgain The integration. umerical 0 : ) )   Γ 0 eae distributions Related 3 · EIBULL . distribution ( . , · ) seteexcursus the (see EIBULL distribution ATRA (3.107d) (3.107b) distribu- (3.106c) (3.107c) (3.107a) (3.107e) (3.107f) has — /D EY Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC ihnndcesn function non–decreasing with .TegnrlCDF general The 3.3.8.1). Sect. thssm eebac oaW a to resemblance some has It h upr of support The /8 2 Fig. In ie tagtln.TeS The line. straight a gives rnfraino h ovninltreprmtrW three–parameter conventional the of transformation ino h W the of tion satisfying where ilstecnetoa he–aaee W three–parameter conventional the yields o fteW the of nor u h pseudo–W the But W the of Modifications 3.3 netn,ie,sligfor solving i.e., Inverting, hc et pnagnrlzto ftepretl functio percentile the of generalization a upon rests which ftecnetoa he–aaee W three–parameter conventional the of S F RIVASTAVA REIMER 37 • • • e loteetnin ie n(.8)–(3.79b). – (3.78a) in given extensions the also See > d d < d w 0 = ( x ehv rsne h W the presented have we ta.(99 eeoe htte alan call they what developed (1989) al. et 0 0 ) F gives gives gives samntnclyicesn ucindpnigo n rm or one on depending function increasing monotonically a is EIBULL EIBULL ( 18)clshsseilvrina version special his calls (1989) u F | u ( d ln u EIBULL P − ∞ − ∞ − = )  | = d distribution. < u < d rnfrain(3.110): transformation − )         eed on depends ln x < u < < u < lim EIBULL → P Ψ( ln  LYMEN d ln 1 1 itiuini eeaiainnihro h am distr gamma the of neither generalization a is distribution 1 0  n − − =  w x −  Ψ( − − = ) ( ∞ exp exp F − x F ∞ − ln ln(1 distribution x = ) ( , ( d /L u ln(1 EIBULL ) . y     F , EIBULL ) d 1 where | ACHENBRUCH , − − x ( −∞ ,b c b, a, − − sfollows: as ie h CDF the gives x − − EIBULL  b e 1 = ) F P u + 1 a P

( ) ()=0 = Ψ(0) EIBULL  x  and oe ihtm–eedn aaees(see parameters time–depending with model poaiiyppr nti ae h following the paper this On -probability–paper. ) )   c ) u d 1  − ; /d  distribution: = d e a x = −  lim − →∞ − ∈ Ψ( distribution. α eeaie W generalized 1 18)mdlrssuo generaliza- a upon rests model (1984) c R o x and + w for for ln ) EIBULL ; xeddW extended ( w β ,c> c b, b x for for Ψ( n = ) + 37 d d x ( ∞ c x P 0 6= 0 = ) ∞ ln( 0 CDF = ) , d d = . 0 6= 0 = x . a ∞      EIBULL − + EIBULL . h pca case special The . a b    )  − r parameters ore ln(1 distribution distribution − (3.108b) (3.111a) (3.108a) (3.109) (3.110) ibution P )  187 1 /c . Downloaded By: 10.3.98.104 At: 14:41 27 Sep 2021; For: 9781420087444, chapter3, 10.1201/9781420087444.ch3 © 2009 byTaylor& FrancisGroup, LLC hc smntnclyicesn when increasing monotonically is which h orsodn Ris HR corresponding The etasomto f(.1a sn 311)gives (3.111b) using (3.111a) of Re–transformation tews tmysatwt eraigpr u nly(for finally but part decreasing a with start may it otherwise hydsusdteseilfunction special the discussed They 188 F ( h x ( x | ,β ν β, α, | ,β ν β, α, 1 = ) = ) > β − β 2 exp x 2 w ν (  x ( ( + ν = ) − x 1) + − exp ν ( x x − ν ν " x 1 − − 2
α − 1 ν ν exp x + − + − 2 β ν x ( − − ; α x ν ( ,ν x, − ν ν + 2 1 − − ) ν β 2 x 1) ≥ − x x 0 ν ν . x ν
2 − +2 #) ν ra)i ilincrease. will it great) x  − ; x , ν
) eae distributions Related 3 ≥ , 0 . (3.111d) (3.111b) (3.111c)