A GUIDE TO THE BURR TYPE XII DISTRIBUTIONS

by

Robert N. Rodriguez*

Department of Statistios University of North CaroUna at ChapeZ HiU

Institute of I~meo Series No. 1064

April? 1976

* This research was supported in part by a National Science Foundation Graduate Fellowship and by the U. S. Arffry Research Office under Grant No. DAHC04-74-C-0030. A Guide to the Burr Type XII Distributions by Robert N. Rodriguez*

(1f31~ Diagrams of areas in the 13 2) plane corresponding to the Burr Type XII distributions are presented.

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Some key words: Burr Type XII distribution; beta coefficient diagram; .

* This research ,vas supported in part by a National Science F01.mdation Graduate Fellowship and by the U. S. Amy Research Office under Grant No. DAHC04-74-C-0030 . .e 2

Suppose that Z is a positive with probability density function k PZ(Z) = (l+z)k+l (Z > 0)

where k is a positive parameter. TIlen for c > 0, X = Zl/c has the cumulative distribution function

(x > 0).

X is said to have the Burr distribution of Type XII with parameters c and k. The purpose of this exercise is to develop a diagram in the

(is!, 82) plane corresponding to distributions of Type XII variables (and their powers). The treatment follows the approach used by Jolmson and Kotz [8] for power transfonnations of gamma variables.

·e Burr [1] introduced a system of distributions ii'1 1942 by considering distribution functions F(x) satisfying the differential equation

g£ = A(F) g(x).

This is a generalization of the Pearson equation; the object was to fit a distribution function, rather than a density, to data and then obtain

the density by differentiation. In the special case where A(F) = F(l-F), the solution of Burr's equation is

1 F(x) = l+exp{-G(x)}

where G(x) = I~oo g(t)dt. The Type XII distribution is a particular

fom of this solution and was discussed in some detail by Burr, who noticed .e 3

that the family gives rise to a useful range of values of ) (a3 and (a4). He recomrilended fitting by a method of cumulative moments and presented a short table of a3 and a4 as functions of c and k. Burr Type XII distributions were ftlTther investigated by Hatke [6] in 1949. She presented a diagram of the moment_ ratio ncoverageY9 using

the (a~~ 0) coordinate system due to Craig [5]. It appears that the Type XII family was not studied again until 1968 ~ when Burr [2] and Burr

and Cislak [4] examined the distributions of sample and range. By then the computation of moments had been siraplified by the use of electronic computers. Burr CL'1d Cislak pointed out that the coverage

is actually much greater than that claimed by Hatke. They presented

.e a revised (a; p 0) diagram which also relates the coverage to that of the Pearson system. A similar chart is to be fomd in a short communication by Burr [3] in which c and k values giving corrbinations of a3 and (14 are tabled. However use of coordinates does not provide the clearest exposition of the Type XII family. In particular the upper and lower

bounds are not apparent~ (nor have they been derived analytically).

Ord [9] presented a (Bp 13 2) version of the chart constructed by Burr and Cislak which does not appear to clarify the region of coverage. In

fact~ there is some d..oubt as to the validity of the shape of the lower bound. Finally it should be noted that none of the diagrams published

so far indicates the effect of rodifying c and k. 4

The rth fooment about the origin of X = ZI/c is

(r < ck)

so that the fourth ror;lent is finite if ck > 4. Burr originally used the restrictions c ~ 1 and k ~ 1 in [l]~ possibly since the density function corresponding to X

kcxc-1 f.,,(x) = 1+1 (x > 0) .J\. (l+xc) (

is l~imodal at x = [(c-I)/(ck+I)]l/C if c > 1 and L-shaped if c = 1. In wllat follows k and c are taken to be positive with ck > 4.

The parametric equations for IB:l and 13 Z are: 2 r (k) A -3r(k)A Al+2Ai IS:"=--~--;or-3 ...... Z ,--~ -e 1 [r(k)Az-AiJ3/2

3 2 2 ~ r Ck)A4-4r Ck)A3Al+6rCk)AZAI-3AI 13 ------...,.~-r;;-- 2 - [rCk) AZ-Ai]3/Z

where Aj = rC~l)rCk-t)~ j = 1~Z~3~4. Tables of the moment-ratios "Jere obtained for fixed values of the power parameter 1/c. These nmnerical results extended the table of cx,3 and cx,4 constructed originally by BuTI' in [1] and were used in the preparation of Figures 1 and Z which are the

aTlalogrtes of the diagrams in Jolmson and Kotz [8]. Clearly a (1Sl~ 8Z) coordinate system is the best choice for displaying the results of the calculations.

Figure 1 shows the loci of Cv\3l, 13 2) points for constant c with .e k varylllg. T.~e Type XII Burr distributions occupy a region s~~ped like 5 FIG. 1. VALUES OF (181, S2) FOR THE TYPE XII BURR DISTRIBUTIONS

-1. 5 -1. 0 -0.5 0.5 1.0 1.5

N nonnal R rectangular L logistic EV extreme value E exponential

H H R 2 PEARSON TYPE I

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Po. I ...,- I " 1 I ~' " ~ I , ~ /:;'1 , fpl ,,/ E', l I ~, I 'I , 6 t'I i/ i/I ~, PEARSON TYPE IV

I'I I .e ,I 8 k=l 6

FIG. 2. EXISTENCE REGIONS FOR THE TYPE XII BURR DISTRIBUTIONS

-1. 5 -1. 0 -0.5 0.5 1.0 1.5

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Ruled sector indicates region 6 of single occupancy; cross- hatched sector indicates region of double occupancy.

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the prow of an ancient Roman galley. As 1< varies from 4/c to +00 the c-constant curve moves in a counter-clockwise direction toward the

tip of the !9prow". For c < 3.6 the c-constant lines tenninate at endpoints with positive 1r31. For c > 3.6 the c-constant lines tenninate at endpoints with negative lSI. These endpoints fonn the lower boundt of the Type XII region. Their parametric equations are rC~1)-3r(~1)r(hl)+2r3cl..l) lim IS:"" - c c c c k~ 1 - [rcb. )-r2c1. )]3!2 cl c 1 rc!t.l ) -4r C~l) rcl..l)+6rC~1)r 2ChI)-3r4ChI) lim 8 = c c c c c c. k~ 2 [rct.l)-r2chl)]2 c c These limits are easily obtained from the parametric equations for Ial

and 13 2 by applying Stirling's fonnula. The lower bound for the Type XII region fonns part of the Weibull curve in the CI61? 13 2) plane. Values for the moment ratios of the Weibull distributions are tabulated

in Chapter 20 of Johnson and Kotz [7]. 82 has a minimum value of about 2. 71 when c; 3.35. The Weibu11 curve passes through the exponential

point CISI = 2? 82 = 9)? which is the endpoint for the Burr curve with c = 1. The Weibull density ftmctions are tmimoda1; for 0 < c ~ 1 the

is at zero, and for c > 1 the mode is located to the right of the origin. The identification of the lower bound with the Weibu11 family can

t We are following the convention of presenting moment ratio diagrams upside-down. Thus the upper bounds in Figures 1 and 2 are referred to as 1I1ower bounds" and conversely. .e 8

be explained as follows: c PiX s (fc)l/C y} = 1 - (1 + t-)-k

= 1 - exp{-k log(l + ~Yc} c c 2 = 1 - exp{ -k[f - ~(t-) + ••• ]} _yC + 1 - e as k + 00.

On the other hand 9 the Type XII distribution can be obtained as a smooth r1ixture of Weibull distributions

(6)0; w>O)

compounded with respect to e~ where e has a standard with parameter k.

-e As c + 00 the endpoints of the c-constant Burr curves approach a point on the Weibull curve. In order to derive the limiting coordinates we use the fact that the ganuna function is analytic in the positive half­ plane, so that

2 r(l+z) = r(l) + zr' (1) + 2rH(l) + ~ r(3) (1) + hr(4) (1) + ... ,

where the coefficients needed are:

tel) = I'

riel) = Wei) = -y 2 2 2 TI r"(l) = y + l;(2) = y +"6

r(3)(I) = _y3 - 3y~(2)- 2~(3) ,e and 9

Here ~(z) is the digammA function, y = 0.57722 is Euleris constant, and 1;;(z) is the Riemann zeta function. (See Whittaker and Watson [10].) It can be shown that

o 5.35.

The limiting Burr curve c = +00 fonns part of the upper bound for the Burr Type XII region, passing through the logistic point (lSi = 0,

'4It 82 = 4.2) and approaching the Weibull curve asymptotically as k + 0+ . In the Burr region below the curve c = co each point (v'S!' 82) corresponds to a unique pair (c,k) and hence a unique Type XII distribution. (See

Figure 2.) Above the curve c = co there are two Type XII distributions

for each (Ial, 82) point, one with c > 3.6 and one with c < 3.6. The upper bound for the Burr region in the positive Ii3l half-plane presents a special problem. It is clearly not a c-limiting member of the Type XII family. Initially it was thought that this section of the upper bound might represent the moment ratios of some generalization of the ~ and several known generalizations were considered

i1 unsuccessfully. For example, a natural licandidate is the Burr Type II y family with distribution function (e- + l)-k. Tabulation of moment ratios shows that the Type II curve lies well within the Type XII region. 10

Actually the upper botmd corresponds to the Burr Type XII distributions for which Ie = 1 and c > 4. This can be shown by applying the method of Lagrange multipliers in order to minimize ~ subject to the constraint

~ ~ 13 2 = constant 4.2. Using the parametric equations for lSI and 132 one obtains a/3l ] [a/32] -1 = [a/3l ][ d/3Z ]-1 [ dC ac aK ak .

This is an extremely complicated equation~ and the algebraic details are omitted~ but it can be shown that k = 1 necessarily. (Undoubtedly a simpler theoretical deriv.ation can be given~ although the result is substantiated numerically.) Consequently the upper bound for IS:l > 0 corresponds to the distributions

P{X$x} = 1 _ ___1_ (x>O; c>4) ~ l+xc

where c = 00 corresponds to the logistic distribution. As indicated by Figure 1 the Burr Type XII region covers sections of the areas corresponding to the main Pearson Types I ~ IV ~ and VI. The section of the Type III line joining the nomal and exponential points

is also included~ as are those Type VII distributions represented by the line segment joining the normal and logistic points. Part of the Type II line segrn.ent is included in the Burr region, but not the rectangular

~ endpoint (1i31 = 0 13 2 = 1.8) as claimed by Burr and Cislak [4]. On the other hand~ the extreme value point is included.

The Burr region contains points in the heterotypic region. The lognormal curve (omitted for simplicity) lies between the Type III and .e 11

Type V curves? so it is clear that the Burr region covers areas corresponding to each of the three families in the Jolmson system.

ACKNOHlEDGEMENT

The author wishes to thank Professor Nonnan Johnson for suggesting .~ this problem. 12

REFERHJCES

[1] Burr, loW. (1942). Cumulative frequency functions. Annals of Mathematiaal Statistias$ 13, 215-232. [2] Burr, I.W. (1968). On a general system of distributions III. The sample range. Journal of the Ame~iaan Statistiaal Assoaiation$ 63, 636-643. [3] Burr, I.W. (1973). Parameters for a general system of distributions to match a grid of as and a~. Communiaations in Statistias, 2, 1-21. [4] Burr, I.W. and Cislak, P.J. (1968). On a general system of distributions, I. Its curve-shape characteristics, II. The sample median. Journal of the Ameriaan Statistical Association, 63, 627-635. [5] Cra,ig, C.C. (1936). A new exposition and chart for the Pearson system of frequency curves. Annals of Mathematiaal Statistias, ?, 16-28. [6] Hatke, M.A. (1949). A certain cumulative probability ftmction. Annals of Mathematiaal Statistias$ 20, 461-463. [7] Jolmson, N.L. and Kotz, S. (1970). Continuous Univariate Dist~ibutions. 2 vo1s., Boston: Houghton Mifflin Company. ·e [8] Jolmson, N.L. and Kotz, S. (1972). Power transformations of gamma variables. Biometrika, 59, 226-229. [9] Ord, J.K. (1972). Families of Frequenay Distributions. New York: Hafner Publishing Company. [10] ~lliittaker, E.T. and Watson, G.N. (1927). A Co~se of MOdern Analysis. London: Cambridge University Press.

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