STATISTICAL PROPERTIES FOR THE GENERALIZED COMPOUND GAMMA DISTRIBUTION

Transcription

1 STATISTICAL PROPERTIES FOR THE GENERALIZED COMPOUND GAMMA DISTRIBUTION SY Mead, M. L' Adratt In this paper, we introduce a general form of compound gamma distriution with four parameters ( or a generalized eta two distriution (GB2)) and some of its statistical properties are investigated. Some special cases and related distriutions of this model are derived. Key words: Compound distriutions; Moments; Beta and incomplete eta functions; Generalized eta two distriution. I. Introduction The theory of mixture distriutions is well known and frequently used in various scientific disciplines. In particular, it has useful applications in industrial reliaility and medical survivorship analysis. The gamma distriution is one of the most common distriutions that plays an important role in statistics. The standard form of the compound gamma distriution (or the standard form of the eta II distriution) was first presented y Duey (1970). The prolem of otaining the maximum likelihood estimators of the 3-parameter compound gamma distriution (or 3-parameter eta II distriution) investigated y El-Helawy et. al. (2002). The properties of the compound Weiull-gamma distriution (or generlized Bun - XII distriution) are derived y Mead (2006). The ojective of this paper is to find a general form of the compound gamma distriution (or a new type of generalized eta II distriution), also to derive and study the characteristics of this model. The suggested distriution can e specialized Department of Statistics, Faculty of Commerce Zagazis University, Egypt. 1

2 to several important distriutions and it also includes some generalized distriutions 2. Generalized Compound Gamma Distriution The proaility density function (pdf) of the four-parameter generalized compound gamma distriution (or generalized eta II distriution GB2) can e otained y compounding the gamma distriution in the form Ax; a, q, 31 4 x - Ar l e - q ( " ), 0 < A < x < co, a,q > 0 F(a) with the gamma distriution in the form 0.4 q, q >0,,0 > 0 g(q;,0)= q0 e integrating over q, the resulting compound density function has the following form x-at -1 f(x;a,o,a,)= ( ma,0)(--t rr a +0) 0<2<x<co, a,o,> 0 where a and B are the shape parameters, a is the location parameter, is the scale parameter and 13(.,.) is well known eta function. The' umulative distriution function (cdf ) of the GM (1) is F(x;a,O,A,)= fic(a,0) ft(a,0) rzlc(a, e) where I c (ct,o) is the incomplete eta ratio, with c -(L the incomplete eta function defined as (1) 2 ) and 13 x (., ) is 2

3 /3 x (a, k) = ix a-1 + xr +k dx 0 The corresponding survival function S(x) and the hazard rate function H(x) are S (x) /3` (a 0) ' /3 (a,0) H(x). 1 x A) a l, r x-a r e) fi c 11 (a,ba ) 1 where, /3 x (a, k) = 13(a, k) fi x (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, =8 SW Os a a Figure (1) 2. Some Properties For The Generalized Compund Gamma Distriution The r th non-centeral moment for the compound density (1) can e derived as 3

4 r 7 ( r ;I t) ( 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 r th centeral moment for 0:2 (1 ) will e frr= (1E1) F(a)110) j. 0 j 0-1 r(a + r nne-r + j), 0>r+ j, r =1,2,3,4,... Using (2), (3) and pdf (1), we can derive the following statistics (3) (i) The mean p and the variance cr 2 of the GB2 (1) are a m (0-1) +1 > 1 and 2 2 a (a + 0-1) a = B> ) (ii) The mode X m of the GB2 (1) will e at the point X m i l+ A 0+1 suject to 82 a,61,1,) f (xl 2 x < 0 i.e. (a _0a >0 3 + Or Ma,0) it is evident that the mode X m 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) 4

5 where c s =( ) and 0 < p <1, the median is otained (4) corresponding P = 0.5. (iv)the mean deviation can e derived as M.D(x) A, f(x ; a,e, 'L) dx [Li z (a +1,0 I3( 1 ) fix (a +LB 1)] a, 0) a 1)kz ( R o rr ea Acr, 0 )(0 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) 4a 2 (a +1Xa +2) + 6 a 3 (a +1) 3a 4 (0-1X0-2X ) (0-1) 2 (9-2X0-3) (0-1)3 (9-2) 0-04 l 2 a (a + (0-1)) (0-1) 2 (0-2)]

6 should e noted that the coefficient of skewness a3 and the coefficient of kurtosis a4 of x does not depend on parameters A and. The shape parameters a and B play a very important role in determining the properties of distriution (1). 4. Some Special Cases The compound gamma distriution defined y (1) can e specialized to different known distriutions such as (a) When a =1, the density (1) reduces to the 3-parameter Pareto distriution of the second kind ( or the 3-parameter compound exponential-gamma distriution with pdf fix;1,0,a,). 19 (1+ ix -11)4+1), O<A<x<oo, 0,> 0. j) (5) L For the density (5) we have the following special cases (i) If = kl c and B =1/c then, the resulting density will e the generalized Pareto distriution derived y Ahsanullah (1991) with the pdf 2 41+c -1 ) AX;INC,A., k/c)=1( 1+C 5 - k 1. 0< < x <oo, c,k >0. k and he restricted that 0 < x < - c -1 if c < O. (ii) If = A then, the resulting density will e the standard Pareto distriution 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 distriution ( or 2-Parameter compound exponential-gamma distriution ) with the pdf f (x ;1, 0,0, (0+1) = x> 0, 6,0 > 0. 6

7 () When 2 = 0, the pdf (1) 'reduces to the 3-parameter eta type 11 distriution ( or 3-paramtere compound gamma distriution ) which was presented y El-Helawy et.at (2002) as the form a -I 0) 1 f(x;a,0,0,)= i1 x ) ( x +F]r a, x > 0, a,0,>0. fl(a,0) For this case, set =1 then, the resu ting density is the standard form of the eta II distriution ( or the standard form of the compound gamma distriution ( Duey (1970))) with the form f(x;a,0,0,1)= I x a xr (a+ ) x > 0, a,9 > 0. fl( a,0) (c) When B =1, the pdf (1) reduces to the 3-parameter compound gamma f exponential distriution with the following pdf A,)- a x - A a-i 1+ x - A -( a +1), 0< A< x <00,,a > 0. (d) When =I, the pdf (1) reduces to other type of compound gamma distriution with three parameters ( or other type of eta II. distriution ) 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 Distriutions The four-parameter generalized eta two (1) can e transfonned to several distriutions as follow (1) The compound ganuna distriution (1) can e written as f (x;a,o, A, y, e + 0) (x A r - A+ r (a+e), r(o)r(a 0<2<x<co, a,a, > O. 7

8 Therefor, the pdf (1) elongs to the general form of Pearson type VI distriution with 8 r ,a=02+1, A-=Yi and 2.=192. x A)Xi (2) When y - ( the pdf (1) can e transformed to the following 4- parameter generalized eta II distriution defined y Mc-Donald and Butler (1990) ffrice,0,y,)= ()74-1(1±11,17)-(a+e) 13(a,O). ) Lid (6) For density (6), we note the following y > 0, a,0,7, > O. (i) When r = q, the pdf (6) reduces to the 4-parameter compound generalized gamma with gamma distriution as following 14+0) f(y;a,,y,q)= i9 (Y r a , 13(a,0) y > 0, a,6 1,y,q >O. a I k (ii) Reparameterized a = - k and 0 = q , the density function (6) reduces to the following model otained y Mielke and Johnson (1974) f(y;k,q,,y)=. A (Y) 1 k -1 ( ] ft(kly, q +1- kl y) y > 0, 1 -(q+1) k,q,,y >O. using the restriction q = kly, Johnson etal. (1994) otained the following pdf 8

9 ftv;k,,71= t vt c ' i f r >, k,,y>0. They called it a Mel ke eta-kaprt distriution and applied it to stream flow and precipitation data. (3) If y = (-v-1[ 2-11 )Y a = - 1 and 0 = 11 then GB2 (1) can e 2 2 transformed to the following generalized F-distriution v2 given y Malik (1967) f iv; y,).,v1,, ir112y)-1,r kvl /v2) 2, \in - Ev2)/ ' (7) 13(9 /2, v2 /2 ) (11+ /v2][y ] /22 y > 0, y, > 0 When y = =1, the density (7) reduces to the following ordinary F- distriution (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 eta II distriution (I) can e transformed to the following 4-parameter compound Weiullgamma model otained y Mead (2006) AY ;0, Yilt,)= r t9 (x- ay ( 14-(x - ily ) ( +1) (8) 0<2<x<oo, 0,y, > O. Reparameterized = qr, the pdf (8) reduces to the generalized Burr XII distriution or the generalized Pareto distriution. 9

10 x- A (5) For y = (1+ [, the pdf (1) can e transfonned to the following standard form of the eta distriution ( of the first kind ) with parameters a,9 Ax;a,0)= fi(c 1, 0 ) I y e-1 0- yr -1, 0 < y <1, a,0 > 0. (6) When y=(k[1+{ x- 21]) X + A, and 0 =1, the pdf (1) can e transformed to the following generalized uniform distriution given y Proctor (1987) f(y;y,k,a,a)= y k a (y- Ari - A) Tr i, A<y<A+k 7, y,k,a > O. x- A ) I (7) If y = In ( + A, then the pdf (1) can e transformed to the following 4-parameter generalized logistic distriution type IV which pointed out y Kalfleisch and prentice (1980) _ a rxa) (y;a,0,a,). e )i +e k -(a+e) (9) - co<y<co, a,0, >O. They oserved that types I, II and III generalized logistic distriutions are all specail cases of density (9). When = 1 and A= 0, distriution (9) can e specialized to the standard form of generalized logistic type IV which was studied y Prentice (1976), he concluded that type IV distriution as an 10

11 alternative for modeling inary response data to the usual logistic model, further, the type IV density is referred to as the exponential generalized eta distriution of the second type denoted y EGB2 (Johnson et. al. (1994)). (8) If y _k Y L r i 1,a = and 0 =1, the pdf (1) can e transformed to the following other type of generalized Pareto distriution given y Hosking and Wallis (1987) Ax;y,k)= I (1-21 k yjr 0<x <, y,k>0. (10) T k When y = 0 and y =1, the pdf (10) reduces to the exponential distriution with mean k and the uniform distriution on (0, k) respectively (Johnson et.al. (1994)). (9) The 3-parameter Weiull distriution can e otained from pdf (1) y considering the following transformation. If Y = In [1+ ILL A -11) + A, a =1 then, Y has the following pdf Ay;0,,.0=0(y-Ar -l e -e(y-a t, 0<,1,< x <co, 6,0 >0. For the four-parameter generalized compound gamma distriution (i) If a =1, the pdf (1) gives the three-parameter Pareto distriution. (ii) If A = 0, the pdf (1) gives the three-parameter eta II distriution. These distriutions have een applied to various economic prolems, life testing and reliaility. 11

12 References 1.Ahsanullah, M. (1991). "On Record Values From the Generalized Pareto Distriution". Paldstan Journal of Statistics, Series A, 7, Duey, S. D. (1970). "Compound Gamma, Beta and F Distriution". Metrika, 16, El-Helawy, A., EI-Gohary, M., and Kot, N. (2002). "Estimation of the Parameters of the Compound Gamma Distriution". The First Conference on Statistics and Commercial and Economic Applications. Faculty of Commerce and Business Administration, Helwan University, Egypt, (1), Hosking, I. R. M., and Wallis, J. R. (1987). "Parameter and Quantile Estimation for the Generalized Pareto Distriution". Technometrics, 29, Johnson, N., Kotz, S., and Balakrishnan, N. (1994). "Continuous Univariate Distriutions-I". Second edition. John Wiley and Sons, New York. 6. Kalfleisch, J. D., and Prentice, R. L. (1980). "The Statistical Analysis of Failure Time Data". New York; Wiley. 7. Malik, H.J. (1967). "Exact Distriution of the Quotient of Independent Generalized Gamma Variales". Canadian Mathematical Bulletin, 10, Mc-Donald, J. B., and Butler, R. J. (1990). "Regression Models for Postitive Random Variales". Journal of Econometrics, 43, Mead, M. E. (2006). "Properties of Compound Weiull-Gamma Distriution". Journal of Commerce Research. Faculty of Commerce, Zagazig University, Egypt, 28, (2),