Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of
10-1-2015 The Kumaraswamy Marshal-Olkin Family of Distributions Morad Alizadeh Gulf University-Bushehr
M. H. Tahir Islamia University, Bahawalpur
Gauss M. Cordeiro Universidade Federal de Pernambuco
M. Mansoor University of Bahawalpur-Pakistan
Muhammad Zubair Government Degree College Kahrorpacca
See next page for additional authors
Published version. Journal of the Egyptian Mathematical Society, Vol. 23, No. 3 (October 2015): 546-557. DOI. © 2015 Elsevier. Used with permission. Authors Morad Alizadeh, M. H. Tahir, Gauss M. Cordeiro, M. Mansoor, Muhammad Zubair, and Gholamhossein Hamedani
This article is available at e-Publications@Marquette: https://epublications.marquette.edu/mscs_fac/406 Journal of the Egyptian Mathematical Society (2015) 23, 546–557
Egyptian Mathematical Society Journal of the Egyptian Mathematical Society
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ORIGINAL ARTICLE The Kumaraswamy Marshal-Olkin family of distributions
Morad Alizadeh a, M.H. Tahir b,*, Gauss M. Cordeiro c, M. Mansoor b, M. Zubair d, G.G. Hamedani e a Department of Statistics, Ferdowsi University of Mashhad, Mashhad, P.O. Box. 91775-1159, Iran b Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan c Department of Statistics, Federal University of Pernambuco, 50740-540 Recife, PE, Brazil d Department of Statistics, Government Degree College Khairpur Tamewali, Pakistan e Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, USA
Received 5 August 2014; revised 30 November 2014; accepted 18 December 2014 Available online 24 February 2015
KEYWORDS Abstract We introduce a new family of continuous distributions called the Kumaraswamy Characterization; Marshal-Olkin generalized family of distributions. We study some mathematical properties of this Estimation; family. Its density function is symmetrical, left-skewed, right-skewed and reversed-J shaped, and Fre´chet distribution; has constant, increasing, decreasing, upside-down bathtub, bathtub and S-shaped hazard rate. We Kumaraswamy distribution; present some special models and investigate the asymptotics and shapes of the family. We derive Marshall-Olkin family; a power series for the quantile function and obtain explicit expressions for the moments, generating Weibull distribution function, mean deviations, two types of entropies and order statistics. Some useful characterizations of the family are also proposed. The method of maximum likelihood is used to estimate the model parameters. We illustrate the importance of the family by means of two applications to real data sets.
2000 MATHEMATICS SUBJECT CLASSIFICATION: 60E05; 62E15; 62N05
ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction
There has be an increased interest in defining new generators or generalized classes of univariate continuous distributions by * Corresponding author. Tel.: +92 62 9255454 (Office). E-mail addresses: [email protected], [email protected] introducing additional shape parameter(s) to a baseline model. (M.H. Tahir). The extended distributions have attracted several statisticians to Peer review under responsibility of Egyptian Mathematical Society. develop new models because the computational and analytical facilities available in programming softwares such as R, Maple and Mathematica can easily tackle the problems involved in computing special functions in these extended distributions. Production and hosting by Elsevier Several mathematical properties of the extended distributions http://dx.doi.org/10.1016/j.joems.2014.12.002 1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. The KwMO family of distributions 547 may easily be explored using mixture forms of the exponentiat- Eq. (4) provides a wider family of continuous distributions. ed-G (‘‘exp-G’’ for short) distributions. The addition of para- It includes the Kw-G family of distributions, the proportional meters has been proved useful in exploring skewness and tail and reversed hazard rate models, the MO family of distribu- properties, and also for improving the goodness-of-fit of the tions and other sub-families. In Table 1, we provide some spe- generated family. The well-known generators are the following: cial models of the KwMO family of distributions. beta-G by Eugene et al. [1] and Jones [2], Kumaraswamy-G The density function corresponding to (4) is given by (Kw-G) by Cordeiro and de Castro [3], McDonald-G (Mc-G) abð1 pÞgðx;nÞGðx;nÞa 1 Gðx;nÞ a b 1 by Alexander et al. [4], gamma-G type 1 by Zografos and fðxÞ¼fðx;a;b;p;nÞ¼ a 1 1 : ð5Þ 1 pG x;n þ 1 pGðx;nÞ Balakrishanan [5], and Amini et al. [6], gamma-G type 2 by Ris- ð Þ tic´and Balakrishanan [7], and Amini et al. [6], odd-gamma-G Eq. (5) will be most tractable when the cdf GðxÞ and the pdf type 3 by Torabi and Montazari [8], logistic-G by Torabi and gðxÞ have simple analytic expressions. Hereafter, a random Montazari [9], transformed-transformer (T-X) (Weibull-X and variable X with density function (5) is denoted by gamma-X) by Alzaatreh et al. [10], discrete T-X by Alzaatreh X KwMO Gða; b; p; nÞ. Further, we omit sometimes the et al. [11], exponentiated T-X by Alzaghal et al. [12], odd expo- dependence on the vector n of the parameters and write simply nentiated generalized (odd exp-G) by Cordeiro et al. [13], odd GðxÞ¼Gðx; nÞ; FðxÞ¼Fðx; a; b; p; nÞ and so on. Weibull-G by Bourguignon et al. [14], exponentiated half-logis- The hrf of X becomes tic by Cordeiro et al. [15], T-X{Y}-quantile based approach by a 1 Aljarrah et al. [16] and T-R{Y} by Alzaatreh et al. [17], Lomax- abð1 pÞgðx; nÞGðx; nÞ hðx; a; b; p; nÞ¼ a a : G by Cordeiro et al. [18], logistic-X by Tahir et al. [19] and new 1 pGðx; nÞ 1 pGðx; nÞ Gðx; nÞ Weibull-G by Tahir et al. [20]. ð6Þ Marshall and Olkin [21] proposed a flexible semi-paramet- ric family of distributions and defined a new survival function This paper is organized as follows. Four special cases of this family are presented in Section 2. Some mathematical proper- G ðxÞ by introducing an additional parameter p such that MO ties are provided in Section 3 such as the shapes of the density p ¼ 1 p and p > 0. They called the parameter p the tilt para- and hazard rate functions, useful expansions for the cdf, pdf meter and interpreted p in terms of the behavior of the hazard and quantile function (qf), explicit expressions for the rate function (hrf) of G x . Their ratio is increasing in x for MOð Þ moments, generating function, mean deviations, Re´nyi and p P 1 and decreasing in x for p 0; 1 . 2ð Þ Shannon entropies and order statistics. Section 4 refers to For any arbitrary continuous probability density function some characterizations of the KwMO family. Estimation of (pdf) g x; n and cumulative distribution function (cdf) ð Þ the model parameters by maximum likelihood is performed G x; n , the cdf and pdf of the Marshall-Olkin (MO) family ð Þ in Section 5. Two applications to real data sets illustrate the of distributions are defined by (x ; p > 0 and p 1 p ). 2 R ¼ potentiality of the new family in Section 6. The paper is con- Gðx; nÞ cluded in Section 7. GMOðxÞ¼ ð1Þ 1 pGðx; nÞ 2. Special models and
ð1 pÞgðx; nÞ Here, we provide a few examples of the KwMO-G family of gMOðxÞ¼ 2 ; x 2 R; 1 pGðx; nÞ distributions. respectively. For p ¼ 0, we have GMOðxÞ¼Gðx; nÞ. 2.1. The KwMO-Exponential (KwMO-E) distribution For a baseline random variable having pdf gðxÞ and cdf GðxÞ, Cordeiro and de Castro [3] defined the two-parameter Kw-G cdf by Let the parent distribution be exponential with parameter k x k x a b k > 0; gðx; kÞ¼ke ; x > 0 and Gðx; kÞ¼1 e . Then, FðxÞ¼1 fg1 GðxÞ : ð2Þ the pdf of the KwMO-E (for x > 0) model is given by
The pdf corresponding to (2) becomes a 1 a b 1 abke kxð1 pÞð1 e kxÞ 1 e kx a 1 a b 1 fKwMOEðxÞ¼ 1 : fðxÞ¼abgðxÞGðxÞ f1 GðxÞ g ; ð3Þ ð1 pe kxÞaþ1 1 pe kx where gðxÞ¼dGðxÞ=dx and a > 0 and b > 0 are two extra shape parameters whose role is to govern skewness and tail weights. 2.2. The KwMO-Lomax (KwMO-L) distribution Now, we propose a new extension of the MO family for a given baseline distribution with cdf Gðx; nÞ, survival function Consider the parent Lomax distribution with positive para- Gðx; nÞ¼1 Gðx; nÞ and pdf gðx; nÞ depending on a para- meters a and b and pdf and cdf given by gðx; a; bÞ¼ðÞa=b meter vector n. Inserting (1) in (2), we define the cdf of the ½ 1 þ ðÞx=b ðaþ1Þ; x > 0, Gðx; a; bÞ¼1 ½ 1 þ ðÞx=b a. Then, new Kumaraswamy Marshal-Olkin (‘‘KwMO’’) family of dis- the pdf of the KwMO-L distribution reduces to tributions by ðaþ1Þ a a 1 aba½ 1 þ ðÞx=b ð1 pÞfg1 ½ 1 þ ðÞx=b a b fKwMOLðxÞ¼ Gðx; nÞ bfg1 p½ 1 þ ðÞx=b a aþ1 FðxÞ¼Fðx; a; b; p; nÞ¼1 1 ; ð4Þ 1 p Gðx; nÞ a a b 1 1 ½ 1 þ ðÞx=b 1 : where a > 0; b > 0 and p > 0 are three additional shape para- 1 p½ 1 þ ðÞx=b a meters. For each baseline G, the ‘‘KwMO-G’’ cdf is given by (4). 548 M. Alizadeh et al.
Table 1 Some special models. S. no. a bpGðxÞ Reduced model 1––0GðxÞ The Kw-G family of distributions [3] 211–GðxÞ The MO family of distributions [21] 311–GðxÞ The exponentiated MO family of distributions [22] 41–0GðxÞ The proportional reversed hazard rate model [23] 5–10GðxÞ The proportional hazard rate model [24] 6110GðxÞ GðxÞ
a 2.3. The KwMO-Weibull (KwMO-W) distribution bGðxÞ FðxÞ as GðxÞ!0; ð1 pÞa Consider the parent Weibull distribution with positive para- abgðxÞGðxÞa 1 fðxÞ as GðxÞ!0; meters k and b. Then, the pdf and cdf are given by ð1 pÞa b 1 k xb k xb gðxÞ¼kbx e and GðxÞ¼1 e , respectively. Then, abgðxÞGðxÞa 1 the pdf of KwMO-W distribution becomes hðxÞ a as GðxÞ!0: ð1 pÞ no a 1 b b abkbxb 1 e k x ð1 pÞ 1 e k x Proposition 2. The asymptotics of Eqs. (4)–(6) as x !1are fKwMOWðxÞ¼ aþ1 given by 1 pe k xb ()"#b 1 b a b 1 e k x 1 FðxÞ að1 pÞGðxÞ as x !1; 1 : k xb 1 pe fðxÞ ba½ ð1 pÞ b gðxÞGðxÞb 1 as x !1; bgðxÞ hðxÞ as x !1: For b ¼ 2, we obtain as special case the KwMO-Rayleigh GðxÞ (KwMO-R) distribution. The shapes of the density and hazard rate functions are described analytically. The critical points of the density of 2.4. The KwMO-Fre´chet (KwMO-Fr) distribution the KwMO-G model are the roots of the equation:
Now, suppose the parent Fre´chet distribution with pdf and cdf g0ðxÞ gðxÞ gðxÞ k k k þða 1Þ pða þ 1Þ given by gðxÞ¼kd x ðkþ1Þ e ðÞd=x and GðxÞ¼e ðÞd= ; x > 0, gðxÞ GðxÞ 1 pGðxÞ respectively, then the pdf of the KwMO-Fr model reduces to a 1 gð xÞGðxÞ hi ¼ að1 bÞ a a : ð7Þ k k a 1 1 pGðx; nÞ 1 pGðx; nÞ GðxÞ abð1 pÞkdk x ðkþ1Þ e ðÞd=x e ðÞd=x nohi 2 fKwMOFrðx;a;b;p;k;dÞ¼ aþ1 d log½fðxÞ k There may be more than one roots to (7). Let kðxÞ¼ . 1 p 1 e ðÞd=x dx2 8 2 3 9 We have a b 1 < =x k = 4 e ðÞd 5 g00ðxÞgðxÞ g0ðxÞ2 g0ðxÞGðxÞ gðxÞ2 1 : k x a 1 : k ; ð Þ¼ 2 þð Þ 2 1 p 1 e ðÞd=x gðxÞ GðxÞ g0ðxÞ 1 pGðxÞ þ pgðxÞ2 pða þ 1Þ 2 Figs. 1 and 2 display some plots of the pdf and hrf of the 1 pGðxÞ KwMO-E, KwMO-L, KwMO-W and KwMO-Fr distributions g0ðxÞGðxÞa 1 aðb 1Þ a for selected parameter values. Fig. 1 indicates that the KwMO 1 pGðxÞ 1 pGðxÞ GðxÞa family generates distributions with various shapes such as sym- 2 a 2 metric, left-skewed, right-skewed and reversed-J. Further, gðxÞ GðxÞ aða 1Þðb 1Þ a Fig. 2 shows that the KwMO family produces flexible hazard 1 pGðxÞ 1 pGðxÞ GðxÞa rate shapes such as constant, increasing, decreasing, bathtub, gðxÞ2GðxÞa 1 upside-down bathtub and S. This indeed reveals that the þ paðb 1Þ 2 a 1 pGðx;nÞ 1 pGðx;nÞ GðxÞa KwMO family is very useful in fitting different data sets with no a 1 various shapes. gðxÞGðxÞa 1 p 1 pGðx;nÞ GðxÞa 1 2 a ðb 1Þ a 2 : 1 pGðx;nÞ 1 pGðx;nÞ GðxÞa 3. Mathematical properties
If x ¼ x0 is a root of (7) then it corresponds to a local maxi- 3.1. Asymptotics and shapes mum if kðxÞ > 0 for all x < x0 and kðxÞ < 0 for all x > x0.It corresponds to a local minimum if kðxÞ < 0 for all x < x0 Proposition 1. The asymptotics of Eqs. (4)–(6) as GðxÞ!0 are and kðxÞ > 0 for all x > x0. It refers to an inflexion point if given by either kðxÞ > 0 for all x – x0 or kðxÞ < 0 for all x – x0. The KwMO family of distributions 549
λ=2.5 α=2 β =5 (a) (b) a=12,b=5,p=0.5 a=0.5,b=2,p=1.5 a=5,b=15,p=1.5 a=1.5,b=55,p=2.5 a=1.5,b=2.5,p=2 a=2.5,b=200,p=5 a=0.5,b=2,p=1.5 a=5,b=100,p=1.5 a=15,b=10,p=1.5 a=3.5,b=150,p=2.5 Density Density 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x x
α=2 β =1.5 δ=1.5 (c) (d) a = 1.5 b = 1p= 0.3 a = 1.5 b = 0.5 p = 0.3 λ =0.5 1.5 a = 2.5 b = 0.5 p = 0.5 a = 2.5 b = 0.5 p = 0.5 λ =0.5 a = 4.5 b = 4p= 0.7 a = 4.5 b = 4p= 0.7 λ =2 a = 120 b = 3.5 p = 0.5 a = 120 b = 3.5 p = 0.5 λ =1.5 a = 15 b = 8.5 p = 0.1 a = 15 b = 8.5 p = 0.1 λ =2 Density Density 0.5 1.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x x
Fig. 1 Density plots: (a) KwMO-E, (b) KwMO-L, (c) KwMO-W and (d) KwMO-Fr models.
The critical point of hðxÞ is the roots of the equation sponds to a local minimum if sðxÞ < 0 for all x < x0 and sðxÞ > 0 for all x > x0. It gives an inflexion point if either g0ðxÞ ða 1ÞgðxÞ pgðxÞ – – þ sðxÞ > 0 for all x x0 or sðxÞ < 0 for all x x0. gðxÞ GðxÞ 1 pGðx; nÞ a 1 p 1 pGðx; nÞ GðxÞa 1 3.2. Useful expansions ¼ agðxÞ a : ð8Þ 1 pGðx; nÞ GðxÞa We can demonstrate that the cdf (4) admits the expansion There may be more than one roots to (8). Let sðxÞ¼ X1 ai 2 2 b G x d log½hðxÞ =dx . We have i ð Þ FðxÞ¼1 ð 1Þ ai : ð9Þ i¼0 i 1 pGðxÞ
00 0 2 0 2 g ðxÞgðxÞ g ðxÞ ða 1Þ½g ðxÞGðxÞ gðxÞ b sðxÞ¼ þ We can obtain an expansion for GðxÞ (b > 0 real non-integer) gðxÞ2 GðxÞ2 as g0ðxÞ 1 pGðxÞ þ pgðxÞ2 X1 p 2 b r 1 pGðxÞ GðxÞ ¼ srðbÞGðxÞ ; ð10Þ r¼0 a 1 a 1 0 p 1 pGðx; nÞ GðxÞ ag ðxÞ a 1 pGðx; nÞ GðxÞa where 2 a 2 a 2 X1 p 1 pGðx; nÞ GðxÞ rþj b j 2 s ðbÞ¼ ð 1Þ : aða 1ÞgðxÞ a a r 1 pGðx; nÞ GðxÞ j¼r j r () a 1 2 p 1 pG x; n G x a 1 ð Þ ð Þ Then, using (10), we obtain agðxÞ a a : 1 pGðx; nÞ GðxÞ P ai 1 k X1 GðxÞ a GðxÞ ¼ Pk¼0 k ¼ c GðxÞk; ð11Þ If x ¼ x0 is a root of (8) then it refers to a local maximum if ai 1 k k 1 pGðxÞ k¼0bk GðxÞ k¼0 sðxÞ > 0 for all x < x0 and sðxÞ < 0 for all x > x0. It corre- 550 M. Alizadeh et al.
λ=1.5 α=2 β =1.5
(a) (b) a=3.5,b=2.0,p=1.5 a=5,b=3.3,p=2.5 a=10,b=5,p=2.0 a=0.5,b=1.7,p=2.5 a=0.4,b=1.9,p=2.0 h (x) h (x)
a=1,b=1,p=0 a=5,b=1.5,p=0.5 a=6,b=1.5,p=0.5 a=0.5,b=1.55,p=1.8 a=0.3,b=1.4,p=1.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x x
β=2.5 δ=1.5
(c) a = 2.5 b = 0.6 p = 1.2 α =1 (d) a = 1.9 b = 0.5 p = 0.1 λ=0.5 a = 3b= 0.35 p = 2.5 α =1.5 a = 0.7 b = 1.3 p = 0.1 λ=0.5 a = 0.9 b = 2.6 p = 3 α =0.5 a = 0.8 b = 2p= 1.5 λ = 2 a = 0.88 b = 2.2 p = 3.1 α =0.5 a = 2b= 0.9 p = 0.3 λ = 1.5 a = 0.85 b = 2p= 3.5 α =0.5 a = 0.2 b = 1.5 p = 3 λ = 2 h (x) h (x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0 012 345 x x
Fig. 2 Hazard plots: (a) KwMO-E, (b) KwMO-L, (c) KwMO-W and (d) KwMO-Fr models. where X1 fðxÞ¼fðx; a; b; p; nÞ¼ bkþ1 hkþ1ðxÞ; ð13Þ X1 jþk ai j k¼0 ak ¼ akða; iÞ¼ ð 1Þ ; j k where h ðxÞ is the exp-G density with power parameter j¼k kþ1 X1 k þ 1. Thus, some mathematical properties of the new distribu- jþk j ai j bk ¼ bkða; p; iÞ¼ ð 1Þ p ; tion can be derived from those properties of the exp-G distri- j k j¼k bution based on (13). For example, the ordinary and and for k 1 incomplete moments and generating function of X can be ! obtained from those quantities of the exp-G distribution. 1 1 Xk c ¼ c ða; p; iÞ¼ a a b ; k k b k b r k r 3.3. Quantile power series 0 0 r¼1 and c0 ¼ a0=b0. Then, we obtain 1 Let QGð Þ ¼ G ð Þ be the baseline qf. The KwMO-G distribu- X1 tion is easily simulated by inverting (4) as follows: if u has a FðxÞ¼Fðx; a; b; p; nÞ¼ b GðxÞk; ð12Þ k uniform Uð0; 1Þ distribution, the solution of the non-linear k¼0 equation P b 0 1 1 i hi1 where ak ¼ i¼0ð 1Þ ckða; p; iÞ; b0 ¼ 1 a0 and, for 1 a i B ð1 pÞ 1 ð1 uÞb C a B C k P 1; bk ¼ ak, and HaðxÞ¼GðxÞ denotes the exponentiat- x ¼ QG@ hi1 A ð14Þ 1 a ed-G (‘‘exp-G’’ for short) cdf with power parameter a > 0. 1 p 1 1 ð1 uÞb The last results hold for real non-integer a. For integer a,it is clear that the indices should stop in integers and we can easi- has the density function (5). ly update the formula. The effects of the shape parameters a and b on the skewness The density function of X can be expressed as an infinite and kurtosis can be considered based on quantile measures. linear combination of exp-G densities, namely Now, we derive a power series for the qf x ¼ QðuÞ¼F 1ðuÞ The KwMO family of distributions 551 of X by expanding (14). First, if QGðuÞ does not have an explic- Eq. (20) is the main result of this section since it allows to it expression, it can usually be expressed as a power series obtain various mathematical quantities for the KwMO family X1 as demonstrated in the next sections. The formulae derived i QGðuÞ¼ ai u ; ð15Þ throughout the paper can be easily handled in most symbolic i¼0 computation software platforms such as Maple, Mathematica and Matlab. where the coefficients ai’s are suitably chosen real numbers which depend on the parameters of the G distribution. For several important distributions, such as the normal, Student 3.4. Moments t, gamma and beta distributions, QGðuÞ does not have explicit expressions but it can be expanded as in Eq. (15). Let Ykþ1(k P 0) be a random variable having the exp-G pdf From now on, we use a result by Gradshteyn and Ryzhik hkþ1ðxÞ with power parameter k þ 1. A first formula for the [25] for a power series raised to a positive integer n (for n 1) nth moment of X follows from (13) as ! n X1 X1 X1 n n n i i EðX Þ¼ bkþ1 EðY Þ: ð21Þ QGðuÞ ¼ ai u ¼ cn;i u ; ð16Þ kþ1 k 0 i¼0 i¼0 ¼ where the coefficients c (for i 1; 2; ...) are determined from Moments of some exp-G distributions are given by Nadarajah n;i ¼ n n and Kotz [26], which can be used to obtain EðX Þ. the recurrence equation (with cn;0 ¼ a ) 0 A second formula for EðXnÞ follows from (21) as Xi 1 X1 c ¼ðia Þ ½mðn þ 1Þ i a c : ð17Þ n;i 0 m n;i m EðXnÞ¼ ðk þ 1Þb sðn; kÞ; ð22Þ m¼1 kþ1 k¼0 Clearly, the quantity cn;i is obtained from cn;0; ...; cn;i 1 and R where sðn; kÞ¼ 1 Q ðuÞn ukdu. then from the quantities a0; ...; ai. 0 G Next, we derive an expansion for the argument of Q ð Þ in The nth incomplete moment of X is determined as G Z (14) y m ðyÞ¼ xn fðxÞdx hi1 n 1 a P 1 ð1 pÞ 1 ð1 uÞb 1 k Z k¼0ak u X1 GðyÞ A P n k ¼ hi1 ¼ 1 k ; 1 a b u ¼ ðk þ 1Þbkþ1 QGðuÞ u du; ð23Þ b k¼0 k 1 p 1 1 ð1 uÞ k¼0 0 where the integral can be computed numerically for most G P 1 i where a ¼ð1 pÞ 1 ð 1Þiþk a b ; b ¼ 1 p and distributions. k i¼0 i k 0 X1 1 i 3.5. Generating function iþkþ1 a b bk ¼ p ð 1Þ : i;k¼0 i k Here, we provide two formulae for the moment generating The ratio of the two power series can be expressed as function (mgf) MðtÞ¼EðetXÞ of X. Clearly, the first one can X1 be expressed from (13) as k A ¼ ck u ; ð18Þ X1 k¼0 MðtÞ¼ bkþ1 Mkþ1ðtÞ; ð24Þ k¼0 where c0 ¼ a0=b0 and the coefficients ck’s (for k P 0) follow from the recurrence equation where Mkþ1ðtÞ is the mgf of Ykþ1. Hence, MðtÞ can be deter- ! mined from the exp-G generating function. A second formula 1 1 Xk c ¼ a b c : for MðtÞ can be derived from (13) as k b k b r k r 0 0 r¼1 X1 MðtÞ¼ ðk þ 1Þbkþ1 qðt; kÞ; ð25Þ Then, the qf of X follows from (14) by combining (15) and (18) i¼0 R as 1 k ! ! where qðt; kÞ¼ exp½tQGðuÞ u du. i 0 X1 X1 X1 So, we can obtain the mgf’s of several distributions directly k k QðuÞ¼QG ck u ¼ ai ck u : ð19Þ from Eqs. (24) and (25). k¼0 i¼0 k¼0 Further, using (16) and (17), we obtain 3.6. Mean deviations
X1 k 0 QðuÞ¼ ek u ; ð20Þ The mean deviations about the mean (d1 ¼ Eðj X l1 jÞ) and k¼0 about the median (d2 ¼ Eðj X M jÞ)ofX are given by P d 2l0 F l0 2m l0 and d l0 2m M , respectively, 1 i 1 ¼ 1 ð 1Þ 1ð 1Þ 2 ¼ 1 1ð Þ where ek ¼ i¼0ai di;k; di;0 ¼ c0 and (for k > 1) 0 where l1 ¼ EðXÞ; M ¼ MedianðXÞ¼Qð0:5Þ is the median, k 0 X Fðl Þ is easily calculated from (4) and m1ðzÞ is the first incom- 1 1 di;k ¼ðkc0Þ ½mði þ 1Þ k cm di;k m: plete moment given by (23) with n ¼ 1. m¼1 552 M. Alizadeh et al.
Now, we provide two alternative ways to compute d1 and After some algebraic manipulations, we obtain d2. First, a general equation for m1ðzÞ can be derived from (13) as Proposition 3. Let X be a random variable with pdf (5). Then, X1 m ðzÞ¼ b J ðzÞ; ð26Þ @ 1 kþ1 kþ1 Efg log½ GðXÞ ¼ abð1 pÞ Aða þ t 1;a þ 1;b 1;p;aÞj ; k¼0 @t t¼0 R @ z E log 1 pG X;n ab 1 p A a 1;a 1 t;b 1;p;a ; where J ðzÞ¼ xh ðxÞdx is the basic quantity to com- ð Þ ¼ ð Þ ð þ Þjt¼0 kþ1 1 kþ1 @t pute the first incomplete moment of the exp-G distribution. GðX;nÞ a @ E log 1 ¼ abð1 pÞ Aða 1;a þ 1;b þ t 1;p;aÞjt¼0; A second general formula for m1ðzÞ can be derived by set- 1 pGðX;nÞ @t ting u ¼ GðxÞ in (13) as where Aða þ t 1; a þ 1; b 1; p; aÞ; Aða 1; a þ 1 t; b 1; X1 p; aÞ and Aða 1; a þ 1; b þ t 1; p; aÞ are defined by Eq. (28). m1ðzÞ¼ ðk þ 1Þbkþ1 TkðzÞ; ð27Þ k¼0 The simplest formula for the entropy of X is given by R GðzÞ k where TkðzÞ¼ 0 QGðuÞu du can be computed numerically. Efg log½fðXÞ ¼ log½abð1 pÞ Efg log½gðX;nÞ Eqs. (26) and (27) may be applied to construct Bonferroni @ þð1 aÞabð1 pÞ Aða þ t 1;a þ 1;b 1;p;aÞj and Lorenz curves defined for a given probability p by @t t¼0 BðpÞ¼m ðqÞ=ðpl0 Þ and LðpÞ¼m ðqÞ=l0 , respectively, where @ 1 1 1 1 þð1 þaÞabð1 pÞ Aða 1;a þ 1 t;b 1;p;aÞj 0 @t t¼0 l1 ¼ EðXÞ and q ¼ QðpÞ is the qf of X at p. @ þð1 bÞabð1 pÞ Aða 1;a þ 1;b þ t 1;p;aÞjt¼0: 3.7. Entropies @t After some algebraic developments, we obtain an alterna- An entropy is a measure of variation or uncertainty of a ran- tive expression for IRðcÞ as dom variable X. Two popular entropy measures are the Re´nyi c 1 and Shannon entropies [27,28]. The Re´nyi entropy of a ran- I ðcÞ¼ log½abð1 pÞ þ R 1 c 1 c dom variable with pdf fðxÞ is defined as () Z X1 1 1 c 1 c log wi;j;kEZk g ½QGðZkÞ ; ð29Þ IRðcÞ¼ log f ðxÞdx ; 1 c 0 i;k¼0 for c > 0 and c – 1. The Shannon entropy of a random vari- where Zk is a beta random variable with parameters k þ 1 and i able X is defined by Efg log½fðXÞ . It is the special case of ð 1Þ c2;k ðb 1Þc one, w ¼ and the Re´nyi entropy when c " 1. Direct calculation yields i;j;k kþ1 i Efg log ½ fðXÞ ¼ log ½ abð1 pÞ Efg log ½ gðX; nÞ X1 cða 1Þþai j a 1 jþk ; þð1 aÞEfg log ½ GðX; nÞ 2;k ¼ ð Þ j¼k j k þð1 þ aÞE log 1 pGðX; nÞ X1 jþk j cða þ 1Þþai j a b ¼ ð 1Þ p and GðX; nÞ 2;k j k þð1 bÞE log 1 : j¼k 1 pGðX; nÞ ! 1 1 Xk c ¼ a a b : 2;k b 2;k b 2;r 2;k r First, we define 2;0 2;0 r¼1 Z a a 1 ua1 u 3 Aða1;a2;a3;p;aÞ¼ a2 1 du 0 ½ 1 pð1 uÞ 1 pð1 uÞ X1 3.8. Order statistics Z a uk X1 1 1;k i a3 k¼0 ¼ ð 1Þ X1 i¼0 i 0 k Order statistics make their appearance in many areas of b1;ku k¼0 ð28Þ Z statistical theory and practice. Suppose X1; ...; Xn is a random X1 1 X1 a3 ¼ ð 1Þi c ukdu sample from X, then the pdf fi:nðxÞ of the ith order statistic, say i 1;k i¼0 0 k¼0 Xi:n, is given by a i 3 X1 ð 1Þ c1;k i Xn i n i ¼ ; j jþi 1 k 1 fi nðxÞ¼K ð 1Þ fðxÞFðxÞ ; i;k¼0 þ : j¼0 j where where K ¼ n!=½ði 1Þ!ðn iÞ! . X1 a1 þ ai j jþk Following Nadarajah et al. [29], the density function of Xi:n a1;k ¼ ð 1Þ ; j k can be given as j¼k X1 jþk j a2 þ ai j X1 b1;k ¼ ð 1Þ p and fi:nðxÞ¼ mr;k hrþkþ1ðxÞ; ð30Þ j¼k j k "# r;k¼0 1 1 Xk c ¼ a a b : 1;k b 1;k b 1;r 1;k r where hrþkþ1ðxÞ denotes the exp-G density function with para- 1;0 1;0 r¼1 meter r þ k þ 1, The KwMO family of distributions 553
Xn i j where the function s is a solution of the differential equation n!ðr þ 1Þði 1Þ!brþ1 ð 1Þ fjþi 1;k R m ¼ ; 0 r;k ðr þ k þ 1Þ ðn i jÞ!j! s0 ¼ g q2 and C is a constant, chosen to make dG ¼ 1. j¼0 gq2 q1 H Clearly, Theorem 1 can be stated in terms of two functions and bk is defined in Eq. (12). Here, the quantities fjþi 1;k are q1 and g by taking q2ðÞ x 1, which will reduce the condi- obtained recursively from f ¼ bjþi 1 and (for k 1) jþi 1;0 0 tion given in Theorem 1 to Eq½ ¼1ðÞjX X P x gðÞx . Howev- er, adding an extra function will give a lot more flexibility, as Xk 1 far as its application is concerned. fjþi 1;k ¼ ðÞkb0 ½mðj þ iÞ k bm fjþi 1;k m: m¼1 Proposition 4. Let X : X ! R be a continuous random nohi Based on Eq. (30), we can easily obtain ordinary and incom- a 1 b variable and let q ðÞ¼x 1 GxðÞ ½ GxðÞ1 a and plete moments and generating function of Xi:n for any parent 2 1 pGxðÞ G distribution. a q1ðÞ¼x q2ðÞx 1 pGxðÞ for x 2 R. The pdf of X is ðÞ5 if and only if the function g defined in Theorem 1 has the form 4. Characterization of the KwMO family 1 a gðÞ¼x 1 þ 1 pGxðÞ ; x 2 R: Characterizations of distributions are important to many 2 researchers in the applied fields. An investigator will be vitally interested to know whether their model fits the Proof. Let X have densityðÞ 5 , then requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution bðÞ1 p a ðÞ1 FxðÞE½ ¼q ðÞjX X P x 1 pGxðÞ 1 ; x 2 R; which provide conditions under which the underlying 2 p distribution is indeed that particular distribution. Various and characterizations of distributions have been established in no many different directions. In this section, several charac- bðÞ1 p 2a ðÞ1 FxðÞE½ ¼q ðÞjX X P x 1 pGxðÞ 1 ; x 2 R; terizations of (KwMO) distribution are presented. These 1 2p characterizations are based on: (i) a simple relationship between two truncated moments; (ii) a single function of and finally the random variable. 1 a – gðÞx q2ðÞ x q1ðÞ¼x q2ðÞx 1 1 pGxðÞ 0 for x 2 R: 4.1. Characterizations based on truncated moments 2 Conversely, if g is given as above, then In this subsection we present characterizations of (KwMO) distribution in terms of a simple relationship between two ðÞaþ1 g0 x q x apgðÞ x 1 pGxðÞ 0 ðÞ2ðÞ truncated moments. Our characterization results presented s ðÞ¼x ¼ a ; x 2 R; here will employ an interesting result due to Gla¨nzel [30] gðÞx q2ðÞ x q1ðÞx 1 1 pGxðÞ (Theorem 1, below). The advantage of the characterizations given here is that, cdf F need not have a closed form and and hence are given in terms of an integral whose integrand depends a sxðÞ¼ ln 1 1 pGxðÞ ; x 2 R: on the solution of a first order differential equation, which can serve as a bridge between probability and differential Now, in view of Theorem 1, X has density functionðÞ 5 . h equation. Corollary 1. Let X : X ! R be a continuous random variable Theorem 1. Let ðÞX; R; P be a given probability space and let and let q2ðÞx be as in Proposition 4. The pdf of X is ðÞ5 if and only H ¼ ½ a; b be an interval for some a < bað ¼ 1; if there exist functions q1 and g defined in Theorem 1 satisfying b ¼1might as well be allowedÞ. Let X : X ! H be a con- the differential equation tinuous random variable with the distribution function F and let q and q be two real functions defined on H such that ðÞaþ1 1 2 0 apgðÞ x 1 pGxðÞ g ðÞx q2ðÞx ¼ a ; x 2 R: gðÞx q2ðÞ x q1ðÞx 1 1 pGxðÞ E½ ¼q1ðÞjX X P x E½ q2ðÞjX X P x gðÞx ; x 2 H; is defined with some real function g. Assume that q ; 1 Remark 1. a The general solution of the differential equation 1 2 ðÞ q2 2 C ðÞH ; g 2 C ðÞH and G are twice continuously differen- in Corollary 1 is tiable and strictly monotone function on the set H. Finally, assume that the equation q g ¼ q has no real solution in the a 1 2 1 gðÞ¼x 1 1 pGxðÞ interior of H. Then G is uniquely determined by the functions Z q q and , particularly ðÞaþ1 1 1 ; 2 g apgðÞ x 1 pGxðÞ ½ q ðÞx q ðÞx dx þ D ; Z 2 1 x 0 g ðÞu GxðÞ¼ C exp ðÞ suðÞdu; a gðÞu q2ðÞ u q1ðÞu where D is a constant. One set of appropriate functions is given 1 in Proposition 4 with D ¼ 2. 554 M. Alizadeh et al.
ðnÞ ðÞb Clearly there are other triplets of functions ðÞq2; q1; g where h ð Þ means the derivative of the function h with respect satisfying the conditions of Theorem 1. We presented one such to n. The observed information matrix can be obtained from triplet in Proposition 4. the authors under request.
4.2. Characterizations based on single function of the random 6. Applications variable In this section, we provide two applications to real data to In this subsection we employ a single function w of X and state illustrate the importance of the KwMO-W and KwMO-Fr dis- characterization results in terms of wðÞX . The following propo- tributions presented in Section 3. The MLEs of the model sition has already appeared in [31] Theorem 2.1.3, so we will parameters are computed and goodness-of-fit statistics for the- just state it here for the sake of completeness. se models are compared with other competing models. The first real data set is a subset of data reported by Bekker e et al. [33] which corresponds to the survival times (in years) Theorem 2. 1 FxðÞ¼½ cwðÞþx d if and only if of a group of patients given chemotherapy treatment alone. 1 d The data consisting of survival times (in years) for 46 patients E½ ¼wðÞjX X P x ewðÞ x ; x 2 ðÞd; 1 ; e þ 1 c are: 0.047, 0.115, 0.121, 0.132, 0.164, 0.197, 0.203, 0.260, 0.282, 0.296, 0.334, 0.395, 0.458, 0.466, 0.501, 0.507, 0.529, 0.534, – where c 0; d; e > 0 are finite constants. 0.540, 0.641, 0.644, 0.696, 0.841, 0.863, 1.099, 1.219, 1.271, 1.326, 1.447, 1.485, 1.553, 1.581, 1.589, 2.178, 2.343, 2.416, hiRemark 2. Taking, e.g., c ¼ 1, d ¼ 1, e ¼ b, wðÞ¼x 2.444, 2.825, 2.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033. a GxðÞ and ðÞ¼d; 1 R, Theorem 2 provides a characteriza- The second real data set was originally reported by Badar 1 pGxðÞ and Priest [34], which represents the strength measured in tion of the cdfðÞ 4 . GPa for single carbon fibers and impregnated at gauge lengths of 1, 10, 20 and 50 mm. Impregnated tows of 100 fibers were 5. Estimation tested at gauge lengths of 20, 50, 150 and 300 mm. Here, we consider the data set of single fibers of 20 mm in gauge with a sample of size 63. The data are: 1.901, 2.132, 2.203, 2.228, Here, we determine the maximum likelihood estimates (MLEs) 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, of the model parameters of the new family from complete sam- 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, ples only. Let x ; ...; x be observed values from the KwMO- 1 n 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, G distribution with parameters a; b; p and n. Let 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, H a; b; p; n > be the r 1 parameter vector. The total ¼ð Þ ð Þ 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, log-likelihood function for H is given by 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, Xn 4.024, 4.027, 4.225, 4.395, 5.020. ‘n ¼ ‘nðHÞ¼n log ½ þabð1 pÞ log ½ gðxi; nÞ In the first application, we shall compare the KwMO-W i¼1 Xn model with other comparative models: the beta-Weibull þða 1Þ log ½ Gðxi; nÞ (BW) [35], the Kumaraswamy-Weibull (KwW) [36], the expo- i¼1 nentiated-Weibull (EW) [37], the Marshall-Olkin extended Xn Weibull (MOW) [38] and the Weibull (W). In the second appli- ða þ 1Þ log 1 pGðxi; nÞ cation, we compare the KwMO-Fr model with other com- i¼1 Xn a parative models: the beta-Fre´chet (BFr) [39], the Gðx ; nÞ þðb 1Þ log 1 i : ð31Þ exponentiated-Fre´chet (EFr) [40], the Marshall-Olkin extend- i¼1 1 pGðxi; nÞ ed Fre´chet (MOFr) [41] and the Fre´chet (Fr). The MLEs are computed using the Limited-Memory Quasi-Newton Code The log-likelihood function can be maximized either directly by for Bound-Constrained Optimization (L-BFGS-B) as well as using the R (AdequecyModel), SAS (PROC NLMIXED) or the the measures of goodness-of-fit including the log-likelihood Ox program (sub-routine MaxBFGS) [32] or by solving the ^ nonlinear likelihood equations obtained by differentiating (31). function evaluated at the MLEs (‘). The measures of good- ness-of-fit including the Akaike information criterion (AIC), The score function UnðHÞ¼ð@‘n=@a;@‘n=@b;@‘n=@p; > consistent Akaike information criterion (CAIC), Bayesian @‘n=@nÞ has components given by hihi information criterion (BIC), Hannan-Quinn information crite- a Gðxi;nÞ Gðxi ;nÞ @‘ n Xn G x ;n Xn log rion (HQIC), Anderson–Darling (A ), Crame´r–von Mises n ð i Þ 1 pGðxi;nÞhi1 pGðxi;nÞ ¼ þ log þð1 bÞ a ; (W ) and Kolmogorov–Smirnov (K–S) statistics are computed @a a 1 pGðxi;nÞ Gðxi ;nÞ i¼1 i¼1 1 * * 1 pGðxi ;nÞ to compare the fitted models. The statistics A and W are Xn a @‘ n Gðx ;nÞ n ¼ þ log 1 i ; described in details in Chen and Balakrishnan [42]. In general, @b b 1 pGðx ;nÞ i¼1 i the smaller the values of these statistics, the better the fit to the Xn @‘ n Gðx ;nÞ n ¼ þða þ 1Þ i ; data. The required computations are carried out using a script @p 1 p i¼1 1 pGðxi;nÞ of the R-language [43]. Xn ðnÞ Xn ðnÞ Xn ðnÞ * @‘n g ðxi;nÞ G ðxi;nÞ G ðxi;nÞ The numerical values of the AIC, CAIC, BIC, HQIC, A , ¼ þða 1Þ þ pða þ 1Þ g x G x * @n i¼1 ð i;nÞ i¼1 ð i;nÞ i¼1 1 pGðxi;nÞ W and K–S statistics are listed in Tables 2 and 4, whereas Xn ðnÞ a 1 Tables 3 and 5 list the MLEs and their corresponding standard G ð xi;nÞGðxi;nÞ það1 pÞ a a ; i¼1 1 pGðxi;nÞ 1 pGðxi;nÞ Gðxi;nÞ errors (in parentheses) of the model parameters. The KwMO family of distributions 555
Table 2 The statistics AIC, CAIC, BIC, HQIC, A*,W* and K–S for the survival times of cancer patients data. Distribution AIC CAIC BIC HQIC A* W* K–S P-value KwMO-W 119.134 120.672 128.167 122.501 0.217 0.027 0.064 0.988 BW 123.995 124.995 131.222 126.689 0.455 0.066 0.098 0.742 KwW 124.189 125.189 131.416 126.884 0.463 0.068 0.099 0.729 EW 122.087 122.673 127.507 124.108 0.470 0.069 0.100 0.717 MOW 121.716 122.301 127.136 123.736 0.444 0.065 0.087 0.858 W 120.247 120.533 123.861 121.594 0.544 0.081 0.109 0.615
Table 3 MLEs and their standard errors (in parentheses) for survival times of cancer patients data. Distribution ab pabk KwMO-W 0.461 0.167 0.003 – 3.256 0.143 (0.412) (0.046) (0.002) – (0.776) (0.173) BW 2.045 4.763 – – 0.691 0.350 (3.287) (21.219) – – (0.633) (1.381) KwW 5.023 21.029 – – 0.302 0.723 (30.532) (144.337) – – (1.288) (3.247) EW – – – 1.612 0.807 1.124 – – – (1.977) (0.527) (1.117) MO-W – – 0.428 – 0.328 1.278 – – (0.182) – (0.282) (0.214) EE – – – – 1.053 0.718 – – – – (0.124) (0.125)
(a) Estimated pdf’s (b)Estimated cdf’s
KwMOW BW KW MOW EW W cdf Density KwMOW BW KW MOW EW W 0.00.20.40.60.8 0.0 0.2 0.4 0.6 0.8 1.0 01234 01234 x x
Fig. 3 Plots of the estimated pdf’s and cdf’s for the KwMO-W, BW, KW, MOW, EW and W models.
Table 4 The statistics AIC, CAIC, BIC, HQIC, A*,W* and K–S for single carbon fibers data. Distribution AIC CAIC BIC HQIC A* W* K–S P-value KwMO-Fr 121.867 122.920 132.583 126.082 0.236 0.042 0.067 0.941 BFr 120.594 121.283 129.166 123.965 0.324 0.061 0.079 0.822 EFr 118.700 119.107 125.130 121.229 0.330 0.061 0.081 0.803 MOFr 119.746 120.153 126.175 122.275 0.394 0.074 0.083 0.782 Fr 121.804 122.004 126.091 123.490 0.642 0.115 0.100 0.553 556 M. Alizadeh et al.
Table 5 MLEs and their standard errors (in parentheses) for single carbon fibers data. Distribution abpakd KwMO-Fr 0.053 1.046 0.0203 – 4.066 6.038 (0.084) (0.901) (0.054) – (1.941) (1.612) BFr 12.547 20.762 – – 1.167 2.937 (91.822) (64.327) – – (1.926) (10.150) EFr – – – 7.031 2.364 4.295 – – – (8.504) (1.027) (1.611) MOFr – – 10.343 – 7.906 2.203 – – (12.421) – (1.142) (0.234) Fr––––5.433 2.721 ––––(0.508) (0.067)
(c) Estimated pdf’s (d) Estimated cdf’s
KwMOFr BFr MOFr EFr Fr cdf Density
KwMOFr BFr MOFr EFr Fr 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0 2345 2345 x x
Fig. 4 Plots of the estimated pdf’s and cdf’s for the KwMO-Fr, BFr, MOFr, EFr and Fr models.
In Table 2, we compare the fits of the KwMO-W model family to real data sets to demonstrate the usefulness of the with the BW, KW, MOW, EW and W models. We note that new family. These special models provide consistently better the KwMO-W model gives the lowest values for the AIC, fits than other competing models. We hope that the proposed BIC, CAIC, HQIC, A*,W* and K–S statistics (for the survival family and its generated models will attract wider applications times of cancer patients data) among the fitted models. So, in several areas such as engineering, survival and lifetime data, the KwMO-W model could be chosen as the best model. The hydrology, economics, among others. histogram of the data and the estimated densities and cdfs are displayed in Fig. 3.InTable 4, we compare the fits of the Acknowledgments KwMO-Fr model with the BFr, MOFr, EFr and Fr models. We note that the KwMO-Fr model gives the lowest values for The authors would like to thank the Editor-in-Chief, the the AIC, BIC, CAIC, HQIC, A*,W* and K–S statistics (for Associate Editor and the two referees for careful reading and single carbon fibers data) among all fitted models. So, the for their comments which greatly improved the paper. KwMO-Fr model can be chosen as the best model. The his- togram of the data and the estimated pdfs and cdfs for the fitted References models are displayed in Fig. 4. It is very clear from Tables 2 and 4,andFigs. 3 and 4 that the KwMO-W and KwMO-Fr models [1] N. Eugene, C. Lee, F. Famoye, Beta-normal distribution and its provide the best fits to the histogram of these data sets. applications, Commun. Stat. Theory Methods 31 (2002) 497– 512. 7. Concluding remarks [2] M.C. Jones, Families of distributions arising from the distributions of order statistics, Test 13 (2004) 1–43. [3] G.M. Cordeiro, M. de Castro, A new family of generalized In this paper, we propose the new Kumaraswamy Marshall- distributions, J. Stat. Comput. Simul. 81 (2011) 883–893. Olkin family of distributions. We study some of its structural [4] C. Alexander, G.M. Cordeiro, E.M.M. Ortega, J.M. Sarabia, properties including an expansion for the density function Generalized beta-generated distributions, Comput. Stat. Data and explicit expressions for the moments, generating function, Anal. 56 (2012) 1880–1897. mean deviations, quantile function and order statistics. The [5] K. Zografos, N. Balakrishnan, On families of beta- and maximum likelihood method is employed for estimating the generalized gamma-generated distributions and associated model parameters. We fit two special models of the proposed inference, Stat. Methodol. 6 (2009) 344–362. The KwMO family of distributions 557
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