Journl of Modern Applied Sttisticl Methods Volume 3 Issue Article 8 5--204 A Compound of Geet Distribution ith Generlized Bet Distribution Adil Rshid University of Kshmir, Sringr, Indi, dilstt@gmil.com T R. Jn University of Kshmir, Sringr, Indi, drtrjn@gmil.com Follo this nd dditionl ors t: http://digitlcommons.yne.edu/jmsm Prt of the Applied Sttistics Commons, Socil nd Behviorl Sciences Commons, nd the Sttisticl Theory Commons Recommended Cittion Rshid, Adil nd Jn, T R. (204) "A Compound of Geet Distribution ith Generlized Bet Distribution," Journl of Modern Applied Sttisticl Methods: Vol. 3 : Iss., Article 8. DOI: 0.22237/jmsm/39897820 Avilble t: http://digitlcommons.yne.edu/jmsm/vol3/iss/8 This Regulr Article is brought to you for free nd open ccess by the Open Access Journls t DigitlCommons@WyneStte. It hs been ccepted for inclusion in Journl of Modern Applied Sttisticl Methods by n uthorized editor of DigitlCommons@WyneStte.
Journl of Modern Applied Sttisticl Methods My 204, Vol. 3, No., 278-286. Copyright 204 JMASM, Inc. ISSN 538 9472 A Compound of Geet Distribution ith Generlized Bet Distribution Adil Rshid University of Kshmir Sringr, Indi T. R. Jn University of Kshmir Sringr, Indi A compound of Geet distribution ith Generlized Bet distribution (GBD) is obtined nd the compound is specilized for different vlues of β. The first order fctoril moments of some specil compound distributions re lso obtined. A chronologicl overvie of recent developments in the compounding of distributions is provided in the introduction. Keyords: Compound distribution, Geet distribution, Generlized Bet Distribution (GBD), fctoril moments Introduction Regrding the problem of the compounding of the probbility distributions, or hs been conducted in this re since 920. It is ell non tht the prmeter in Poisson distribution is considered to be gmm vrite in the fmous rticle by Greenood nd ule (920). Sellm (948) derived probbility distribution from the binomil distribution by regrding the probbility of success s bet vrible beteen sets of trils. The interreltionships mong compound nd generlized distributions ere first explored by Gurlnd (957) fter hich, Molenr (965) discussed some importnt remrs on mixtures of distributions. Dubey (970) derived compound gmm, bet nd F distributions by compounding gmm distribution ith nother gmm distribution nd reducing it to the bet st nd 2 nd ind nd to the F distribution vi suitble trnsformtions. The ppliction of compounding of distributions to clculte moments s explored by Dycz (973). The problem of compounding of distributions s further ddressed by Gerstenorn (993, 996) ho proposed severl compound distributions; Gerstenorn obtined compound of gmm Adil Rshid is PhD Scholr in the Deprtment of Sttistics. Emil t: dilstt@gmil.com. T. R. Jn is Fculty member in the Deprtment of Sttistics. Emil t: drtrjn@gmil.com. 278
RASHID & JAN distribution ith exponentil distribution by treting the prmeter of gmm distribution s n exponentil vrite nd obtined compound of poly ith bet. Gerstenorn (2004) lso found compound of generlized negtive binomil distribution ith generlized bet distribution by treting the prmeter of generlized negtive binomil distribution s generlized bet distribution. Ali, Aslm nd Kzmi (20) improved the informtive prior for the mixture of Lplce distribution under different loss functions. Rshid nd Jn (203) recently obtined compound of zero truncted generlized negtive binomil distribution ith tht of generlized bet distribution. A brod rnge of relevnt references cn be found in studies by Johnson, Kotz nd Kemp (992). The compounding of probbility distributions The folloing definition nd reltions re needed for compounding probbility distributions. A certin compound distribution rises hen ll (or some) prmeters of distribution vry ccording to some probbility distribution, clled the compounding distribution. Suppose Xy is rndom vrible ith distribution function F(x y) tht depends on prmeter y. If prmeter y is considered to be rndom vrible ith distribution function G(y), then the distribution tht hs the distribution function of X is defined by F x cydg y H x () hich is clled compound, here c is n rbitrry constnt or constnt bounded on some intervl (Gurlnd, 957). The occurrence of the constnt c in () hs prcticl justifiction insmuch s the distribution of rndom vrible, in describing phenomenon, often depends on prmeter tht is itself reliztion of nother rndom vrible multiplied by certin constnt. A vrible tht hs distribution function () ill be symbolized by X nd ill be clled compound of the vrible X ith respect to the compounding. Reltion () is symbolized s follos: H x F x cy G y (2) 279
COMPOUND GEETA AND GENERALIZED BETA DISTRIBUTION Consider the cse hen one vrible is discrete ith probbility function P X x cy, if prmeter y is rndom vrible ith density g(y), then () is expressed by i ( ) ( ) ( ) h x P X x g y P X x cy dy (3) i i i Compounding the Geet Distribution ith the Generlized Bet distribution Suppose X is discrete rndom vrible defined over positive integers. The rndom vrible X is sid to hve Geet distribution ith prmeters nd if x x xx ; x,2,... P x; x x 0 ; otherise (4) here 0 nd. The upper limit on hs been imposed for the existence of the men. When, the Geet distribution degenertes nd its probbility mss is concentrted t point x (Consul, 990). The Generlized Bet Distribution (GBD) is distribution given by the density function r y y ;0 y 0; y 0 or y r ;,,, Br /, GB y b r (5) here,,, 0 b r nd ( /, B r ) is bet function. Distribution (5) is specil limit cse of the Bessel distribution (Srod, 973; Seeryn, 986) tht hs been pplied in relibility theory (Oginsi, 979). Consider Geet distribution (4) tht depends on cy : 280
RASHID & JAN x x xx P x; cy cy cy, x,2,3... x x (6) here 0cy, nd is rndom vrible ith GBD (5). Theorem The probbility function of the compound of Geet distribution ith GBD is x x / x r P GB x D c B, K 0 (7) here D x c x x B r /, x x nd x,2,3,,, b,, r 0, 0 cy nd cy Proof: From (3), (5) nd (6) xr2 y xx P GB x D y cy dy 0 K x x xr 2 y K 0 0 D c y dy here D x x c x x. r Br /, Substituting, y t, results in 28
COMPOUND GEETA AND GENERALIZED BETA DISTRIBUTION xr x x P GB x D c t t dt K 0 0 (8) here, x,2,3,,, b,, r 0, nd 0 c. Using the definition of bet function, (7) is obtined. Specil Cses Cse I: When 2 in (4), Hight s distribution results nd compound of the Hight distribution ith generlized bet follos from (8): * x x r P2GB x D 2 c B, 0 (9) here 2x x x c * 2x x D2 ; x,2,3...,, b,, r 0, 0 cy. B r /, Cse II: If b = / nd =, in (5), the bet distribution nd compound of Geet distribution ith bet distribution follo from (8): x c x x here = B r, * x x P Bx D 3 c Bx r, 0 (0) x. Cse III: When β = 2 nd b = /, = in (4) nd (5), respectively obtined re the Hight nd bet distributions nd compound of the Hight distribution ith bet distribution follos from (8): 282
RASHID & JAN here D * 4 x * x P2B x D4 c Bx r, () 0 2x c 2x x B r, x Fctoril moments of the Compound of Geet distribution ith Generlized Bet distribution nd some specil cses Let X y nd X be rndom vrible ith distribution function F ( x y) nd H(X), respectively (see ()), nd let prmeter y hve distribution G(y). Keeping in mind the formul for the so-clled fctoril polynomil l. x x x x x l l l l y m E X E X dg y (2) is clled fctoril moment of order l of the vrible X ith compound distribution (). Reltion (2) is symbolized s l ^ y E X G y. (3) Theorem 2 The first order fctoril moments of the compound of Geet distribution ith GBD is given by m cy GB y b r ; ;,,, 0 c /, B r / r r B, c B, (4) 283
COMPOUND GEETA AND GENERALIZED BETA DISTRIBUTION Proof: by The first order fctoril moments of the Geet distribution is given m,. Thus, from (3), the st order fctoril moment of the compound of the Geet distribution ith Generlized bet distribution if cy is r ;,,,, cy 0 cy y m cy GB y b r y r dy B r, c 0 r y r y y dy c y dy. r B r, 0 0 Substituting, y t results in r r c 0 t t dt c t t dt B r, 0 0 r r c 0 t t dt c t t dt B r, 0 0 (5) Using the definition of bet function, (4) is obtined. Specil Cse When b = / nd = in (5), the st order fctoril moment of the compound of Geet distribution ith bet distribution is obtined s c m ; cy B y;,/, r, Br, cb r,. B r 0, 284
RASHID & JAN Theorem 3 First order fctoril moments of the compound of Hight ith generlized bet distribution. m 2; cy GB y;, b, r, 2 c 0 r r (6) B, c B,. B r, Proof: The result follos directly from (5) for 2, m cy GB y b r 2;,,,, 2c /, / r r 0 B r t t dt c t t dt 0 0 (7) hich yields (6). Specil cse When b=/, = in (7) the folloing result is obtined: 2c 0 m 2; cy B y;,/, r, B r, cb r, B r, hich gives the st order fctoril moment of the compound of the Hight distribution ith bet distribution. References Ali, S., Aslm, M., & Kzmi, S. M. (20). Improved informtive prior for the mixture of LPlce distribution under different loss functions. Journl of Relibility nd Sttisticl Studies, 4(2): 57-82. Consul, P. C. (990). Geet distribution nd its properties. Communictions in Sttistics Theory nd Methods, 9: 305-3068. 285
COMPOUND GEETA AND GENERALIZED BETA DISTRIBUTION Dubey, D. S. (970). Compound gmm, bet nd F distributions. Metri, 6(): 27-3. Dycz, W. (973). Appliction of compounding of distribution to determintion of moments in Polish. Mtemty, 3: 205-230. Gerstenorn, T. (993). A compound of the generlized gmm distribution ith the exponentil one. Recherches surles deformtions, 6(): 5-0. Gerstenorn, T. (996). A compound of the Poly distribution ith the bet one. Rndom Opertors nd Stochstic Equtions, 4(2): 03-0. Gerstenorn, T. (2004). A compound of the generlized negtive binomil distribution ith the generlized bet one. Centrl Europen Journl of Mthemtics, 2(4): 527-537. Greenood, M., & ule, G. U. (920). An inquiry into the nture of frequency distribution representtive of multiple hppenings ith prticulr reference to the occurrence of multiple ttcs of disese or of repeted ccidents. Journl of the Royl Sttisticl Society, 83: 255-279. Gurlnd, J. (957). Some interreltions mong compound nd generlized distributions. Biometri, 44: 265-268. Johnson, N. L., nd Kotz, S. (969). Discrete distributions (First Edition). Boston, MA: Houghton Miffin. Molenr, W. (965). Some remrs on mixtures of distributions. In Proceedings of the 35 th Session of the Interntionl Sttisticl Institute, Belgrde (Bulletin de l'institut interntionl de sttistique, 4), pp. 764-765. Oginsi, L. (979). Appliction of distribution of the Bessel type in relibility theory. Mtemty, 2: 3-42, (in Polish). Rshid, A., & Jn, T. R. (203). A compound of zero truncted generlized negtive binomil distribution ith the generlized bet distribution. Journl of Relibility nd Sttisticl Studies, 6(): -9. Seeryn J. G. (986). Some probbilistic properties of Bessel distribution. Mtemty, 9: 69-87. Sellm, J. G. (948). A probbility distribution derived from the binomil distribution by regrding the probbility of success s vrible beteen the sets of trils. Journl of the Royl Sttisticl Society, Series B, 0: 257-26. Srod, T. (973). On some generlized Bessel- type probbility distribution. Mtemty, 4: 5-3. 286