STATISTICAL PROPERTIES FOR THE GENERALIZED COMPOUND GAMMA DISTRIBUTION

SY Mead, M. L'

Abdratt

In this paper, we introduce a general form of compound gamma distribution with four parameters ( or a generalized beta two distribution (GB2)) and some of its statistical properties are investigated. Some special cases and related distributions of this model are derived.

Key words: Compound distributions; Moments; Beta and incomplete beta functions; Generalized beta two distribution.

I. Introduction

The theory of mixture distributions is well known and frequently used in various scientific disciplines. In particular, it has useful applications in industrial reliability and medical survivorship analysis.

The gamma distribution is one of the most common distributions that plays an important role in statistics. The standard form of the compound

gamma distribution (or the standard form of the beta II distribution) was first presented by Dubey (1970). The problem of obtaining the maximum likelihood estimators of the 3-parameter compound gamma distribution (or 3-parameter beta II distribution) investigated by El-Helbawy et. al. (2002). The properties of the compound Weibull-gamma distribution (or generlized Bun-

XII distribution) are derived by Mead (2006). The objective of this paper is to find a general form of the compound gamma distribution (or a

new type of generalized beta II distribution), also to derive and study the characteristics of this model. The suggested distribution can be specialized

Department of Statistics, Faculty of Commerce Zagazis University, Egypt.

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to several important distributions and it also includes some generalized

distributions

2. Generalized Compound Gamma Distribution

The probability density function (pdf) of the four-parameter

generalized compound gamma distribution (or generalized beta II

distribution GB2) can be obtained by compounding the gamma distribution

in the form

Ax; a, q, 31 4 x - Arl e-q(") , 0 < A < x < co, a,q > 0 F(a)

with the gamma distribution in the form

0 b .4 b —q , q >0, b,0 > 0 g(q;b,0)= — q0 e

integrating over q, the resulting compound density function has the

following form

x-AT -1 ( rra +0) f(x;a,O,A,b)= b ma,0) (--T- 1+ --b--

0<2<x<co, a,O,b> 0

where a and B are the shape parameters, a is the location parameter, b is

the scale parameter and 13(.,.) is well known beta function. The' umulative

distribution function (cdf ) of the G M (1) is

F(x;a,O,A,b)= fic(a,0) ft(a,0)

rzlc(a ,e)

where Ic(ct,O) is the incomplete beta ratio, with c -(L 2) and 13x (.,•) is

the incomplete beta function defined as

(1)

2

SW Os

10 20 30

/3x (a, k) = ixa-1 + xr+k dx 0

The corresponding survival function S(x) and the hazard rate function

H(x) are

/3` (a 0) S (x)— ' /3 (a,0)

H(x). 1 x — A)al, rx-Are)

bfic (a,BA b ) 111 b

where, /3x (a, k) = 13(a, k) — fix (a,k)

Figure (1) illustrate the shape of the pdf (1) and corresponding S (x)

and H(x) for selected values of the unknown parameters, that is, a=5,0=5,2= 0.02 , b =8

a a

Figure (1)

2. Some Properties For The Generalized Compund Gamma Distribution

The rth non-centeral moment for the compound density (1) can be derived as

3

r 7

( r ;I t) (b 1(a + r — j)1(0 — r + j),

(2) /4= lict)110) j=0 0

On+ j, r =1,2,3,4,...

and the corresponding rth centeral moment for 0:2 (1 ) will be

r(a + r — nne-r + j), (3) frr= (1E1)

F(a)110) j.0 j 0-1

0>r+ j, r =1,2,3,4,...

Using (2), (3) and pdf (1), we can derive the folloWing statistics

(i) The mean p and the variance cr 2 of the GB2 (1) are

b a > 1 m (0-1) +1

and

2 b2 a (a + 0 - 1) a = B> 2 - - 2)

(ii) The mode Xm of the GB2 (1) will be at the point

Xm —bi l+ A 0+1

subject to 82

f (xl a,61,1,b)

< 0 i.e. x

2

(a _0a-20+00+4 >0

• b3 + Or +6 7+1 Ma,0)

it is evident that the mode Xm only exists when a > 1.

(iii)The percential estimators, X p is given as the solution of the following

equation

13.(cr, 0)= P fi(c CO) (4)

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where cs =( ) and 0 < p <1, the median is obtained (4)

corresponding P = 0.5. (iv)The mean deviation can be derived as

M .D(x) — f(x ; a,e, 'Lb) dx A,

I3( b [Liz (a +1,0 —1) — fix (a +LB —1)] a, 0) b rr eA

Acr, 0)(0

a 1)kz ( R o

where, z = a ) 0 —1

(v) The coefficient of skewness a3 and the coefficient of kurtosis a4 of the density (1) are given as

P3

2

a(a +1)(a +2) 3a 2 (a +1) + 2a 3 (0 -1)(0 -2)(0 - 3) (0 -1) 2 (0 - 2) (0 - 1?

[

a (a + (0 -0)1 - (0 -1)2 (0 - 2)

0>3.

e, •4 4 -= 2.

P2

—{

a(a +1Xa +2Xa +3) 4a2 (a +1Xa +2) + 6 a 3 (a +1) 3a4 (0 -1X0 -2X0 -30 - 4) (0 -1)2 (9 -2X0 -3) (0 -1)3 (9 -2) 0-04

2

0 4. l a (a + (0 -1)) (0 -1)2 (0 -2)]

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should be noted that the coefficient of skewness a3 and the coefficient of kurtosis a4 of x does not depend on parameters A and b. The shape parameters a and B play a very important role in determining the properties of distribution (1).

4. Some Special Cases

The compound gamma distribution defined by (1) can be specialized to different known distributions such as

(a) When a =1, the density (1) reduces to the 3-parameter Pareto

distribution of the second kind ( or the 3-parameter compound exponential-gamma distribution with pdf

fix;1,0,A,b). 19 (1+ ixb j)

-11)4+1), O<A<x<oo, 0,b> 0 . (5) L

For the density (5) we have the following special cases (i) If b = kl c and B =1/c then, the resulting density will be the generalized

Pareto distribution derived by Ahsanullah (1991) with the pdf

k

2 41+c-1 ) AX;INC,A., k/C)=1(

k 1+C5-1. 0< < x <oo, c,k >0 .

and he restricted that 0 < x < - c-1 if c < O. (ii) If b = A then, the resulting density will be the standard Pareto

distribution of the first kind with the form

f(x ;1,0 , A) = x- (0 x> A, 0> 0 . (iii) When A= 0, the pdf (5) reduces to the 2-parameter Lomax distribution

( or 2-Parameter compound exponential-gamma distribution ) with the pdf

—(0+1) f (x ;1, 0 ,0 ,

b b = (1 +PT x> 0, 6,0 > 0.

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(b) When 2 = 0, the pdf (1) 'reduces to the 3-parameter beta type 11

distribution ( or 3-paramtere compound gamma distribution ) which

was presented by El-Helbawy et.at (2002) as the form

a -I 0) f(x;a,0,0,b)= 1 ix) (1 +Fx ]ra , x > 0, a,0,b>0. b fl(a,0) b b

For this case, set b =1 then, the resu ting density is the standard form of the beta II distribution ( or the standard form of the compound

gamma distribution ( Dubey (1970))) with the form

f(x;a,0,0,1)= I xa -1 0 + xr(a+°) x > 0, a,9 > 0 . fl( a,0)

(c) When B =1, the pdf (1) reduces to the 3-parameter compound gamma

exponential distribution with the following pdf

f A,b)-a x - A a-I 1+

-( x - A

a +1)

, 0< A< x <00, b,a > 0. b b b (d) When b =I, the pdf (1) reduces to other type of compound gamma

distribution with three parameters ( or other type of beta II .

distribution ) as the form

Ax;a,0,2,1). I -1 [1 + - At(a+9) , 13(a,0) O<A<x<co, a,0>0.

5. Some Related Distributions

The four-parameter generalized beta two (1) can be transfonned to several distributions as follow (1) The compound ganuna distribution (1) can be written as

f (x;a,O, A, by, be + 0)

(x A r - A+ br(a+e) , r(o)r(a 0<2<x<co, a ,A,b > O.

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Therefor, the pdf (1) belongs to the general form of Pearson type VI distribution with

8 r- 91 -02-1,a=02+1, A-b=Yi and 2.=192.

x A)Xi (2) When y - b( the pdf (1) can be transformed to the following

4- parameter generalized beta II distribution defined by Mc-Donald and Butler (1990)

ffrice,0,y,b)= ()74-1(1±11,17)-(a+e) b13(a,O)b .) Lid (6)

y > 0, a,0,7,b > O.

For density (6), we note the following

(i) When br = q, the pdf (6) reduces to the 4-parameter compound

generalized gamma with gamma distribution as following

14+0) i9 f(y;a,,y,q)= (Yr a-1 1-1- , y > 0, a,61,y,q >O. 13(a,0) a I

(ii) Reparameterized a = -k

and 0 = q + 1 - -k , the density function (6)

reduces to the following model obtained by Mielke and Johnson (1974)

1 -(q+1) f(y;k,q,b,y)=

bA

.

ft(kly , q +1- kl y)(Y)k -1(1 +11-]

y > 0, k,q,b,y >O.

using the restriction q = kly , Johnson etal. (1994) obtained the following pdf

8

•

vtc' if 11.17 r ftv;k,b,71= t •b

They called it a Mel ke• beta-kaprt distribution and applied it to

stream flow and precipitation data.

(3) If y = b (-v-1[ b

2-1)Y a = -11 and 0 = 11 then GB2 (1) can be 2 2

transformed to the following generalized F-distribution v2

given by Malik (1967)

, ,r f iv; y,b). kvl /v2)

,v1

2 , ibr112y)-1

\in -Ev2)/ /2

' (7) 13(9 /2 , v2 /2 ) (11+ /v2][y b]

, 2

y > 0, y,b > 0

When y = b =1 , the density (7) reduces to the following ordinary F-

distribution (vi+v2)

f(y). (viiv2) y) 2 (i k, 2 ),

fi(v 02, v2/2 ) v2 v2 y > 0

(4) If y = (x - 2)1/7 + A. and a =1, then the generalized beta II distribution

(I) can be transformed to the following 4-parameter compound Weibull-

gamma model obtained by Mead (2006)

AY ;0, Yilt,b)= r t9 (x-

b

aY (14-(x -b

ilY ) (° +1) (8)

0<2<x<oo, 0,y,b > O.

Reparameterized b = qr , the pdf (8) reduces to the generalized Burr XII

distribution or the generalized Pareto distribution.

> , k,b,y>0.

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(5) For y = (1+ [x- A

, the pdf (1) can be transfonned to the following

standard form of the beta distribution ( of the first kind ) with parameters a,9

Ax;a,0)= I y e-1 0- yr -1 , 0 < y <1, a,0 > 0 . fi(c1 , 0)

(6) When y=(k[1+{x- 21]) X

+ A, and 0 =1, the pdf (1) can be

transformed to the following generalized uniform distribution given by Proctor (1987)

f(y;y,k,a,A)= y k a (y- Arib - A) Tri ,

A<y<A+k 7, y,k,a > O.

x- A ) I (7) If y = b In (— + A, then the pdf (1) can be transformed to the

following 4-parameter generalized logistic distribution type IV which pointed out by Kalbfleisch and prentice (1980)

_arxA) -(a+e) (y;a,0,A,b). e )i +e k b

- co<y<co, a,0,b >O. They observed that types I, II and III generalized logistic distributions are all specail cases of density (9). When b = 1 and A= 0, distribution (9) can be specialized to the standard form of generalized logistic type IV which was studied by Prentice (1976), he concluded that type IV distribution as an

(9)

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alternative for modeling binary response data to the usual logistic model, further, the type IV density is referred to as the exponential generalized beta distribution of the second type denoted by EGB2 (Johnson et. al.

(1994)).

(8) If y _k L

ri 1 ,a = — and 0 =1, the pdf (1) can be Y b

transformed to the following other type of generalized Pareto

distribution given by Hosking and Wallis (1987)

Ax;y,k)= I

T (1- 21 yjr 0<x <—

k, y,k>0. (10)

k When y = 0 and y =1, the pdf (10) reduces to the exponential distribution

with mean k and the uniform distribution on (0, k) respectively (Johnson

et.al. (1994)). (9) The 3-parameter Weibull distribution can be obtained from pdf (1) by

considering the following transformation.

If Y = In [1+ ILLA-11)b + A, a =1 then, Y has the following pdfb

Ay;0,b,.0=b0(y-Ar-le-e(Y-At , 0<,1,< x <co, 6,0 >0.

For the four-parameter generalized compound gamma distribution

(i) If a =1, the pdf (1) gives the three-parameter Pareto distribution.

(ii) If A = 0, the pdf (1) gives the three-parameter beta II distribution.

These distributions have been applied to various economic problems,

life testing and reliability.

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References

1.Ahsanullah, M. (1991). "On Record Values From the Generalized Pareto Distribution". Paldstan Journal of Statistics, Series A, 7, 129-136.

2. Dubey, S. D. (1970). "Compound Gamma, Beta and F Distribution". Metrika, 16, 27-31.

3. El-Helbawy, A., EI-Gohary, M., and Kotb, N. (2002). "Estimation of the Parameters of the Compound Gamma Distribution". The First Conference on Statistics and Commercial and Economic Applications. Faculty of Commerce and Business Administration, Helwan University, Egypt, (1), 139-157.

4. Hosking, I. R. M., and Wallis, J. R. (1987). "Parameter and Quantile Estimation for the Generalized Pareto Distribution". Technometrics, 29, 339-349.

5. Johnson, N., Kotz, S., and Balakrishnan, N. (1994). "Continuous Univariate Distributions-I". Second edition. John Wiley and Sons, New York.

6. Kalbfleisch, J. D., and Prentice, R. L. (1980). "The Statistical Analysis of Failure Time Data". New York; Wiley.

7. Malik, H.J. (1967). "Exact Distribution of the Quotient of Independent Generalized Gamma Variables". Canadian Mathematical Bulletin, 10, 463-465.

8. Mc-Donald, J. B., and Butler, R. J. (1990). "Regression Models for Postitive Random Variables". Journal of Econometrics, 43, 227-251.

9. Mead, M. E. (2006). "Properties of Compound Weibull-Gamma Distribution". Journal of Commerce Research. Faculty of Commerce, Zagazig University, Egypt, 28, (2), 23-34.

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