On mixing generalized poison with Generalized Gamma distribution

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1 34 WALFORD I.E. CHUKWU(*) and DEVENDRA GUPTA (**) On mixing genealized poison with Genealized Gamma distibution CONTENTS: Intoduction Mixtue. Refeences. Summay. Riassunto. Key wods (*) Walfod I.E. Chukwu depatment of Statistics, Univesity of Nigeia, Nsukka, Nigeia. (**) Devenda Gupta, Depatment of Mathematics and Compute Science, Nofolk State Univesity, Nofolk VA 3504 USA REFERENCES [] CONSUL,P.C. and JAIN G.C.(973a) A genealization of the poisson distibution Technometics 5, [] GUPTA, G. (984) Two new discete distibutions, Biom. J [3] STACY, E.W. (96) A genealization of the Gamma distibution, Annals of Mathematical Statistics 33, [4] STACY, E. W. and MIHRAN, G.G. (965) Paamete estimation of a genealized Gamma distibution, Technometics 7, SUMMARY Stacy and Mihan (965) intoduced a thee paamete genealization of the gamma distibution which has application in eliability theoy. They obtained some moments of that distibution. Consul and Jain (973a) and Gupta (984) developed the genealized Poisson which also has application in eliability theoy. In this pape we obtain a genealization of Gupta (984) new discete distibution. The mean, vaiance and the thid moment, fouth moment and β and β ae obtained RIASSUNTO Stacy e Mihan (965) hanno poposto una genelizzazione della distibuzione gamma (con te paametic), deteminandone alcuni momenti. Consul a Jain (973a) e Gupta (984) hanno consideato una genealizzazione della legge di Poisson con applicazione di affidabilita. In qesto aticolo gli AA. Pocedono ad una genealizione della nuova distibuzione disceta di Gupta (984) ottenendo anche la media, la vaianza, il tezo e quato memento, β e β. KEY WORDS: Poisson distibution, Gamma distibution, genealized ; mixtue; moments.

2 35 INTRODUCTION The genealized gamma (GG) distibution was developed by Stacy (96) by supplying paamete p as an exponential facto of the gamma distibution. That distibution has thee paametes ν,, ρ in the fom (.) ( ) x exp{- ρ} f(x; ν,,ρ) ρ ν- x ν ν Γ( ) ρ 0 < x <, ρ >0. ν, >0 In the same pape she obtained the distibution of some functions of independent genealized gamma vaiables. Stacy and Mihan (965) Intoduced a futhe genealization (GG) given late. They also discussed estimation of paametes using a modified method of moment technique. In this pape we will be woking with the genealized gamma distibution (GG) of Stacy and Mihan (965) given by f(x; ν,,ρ) - x ρ x exp{- ρ} Γ( ν ) Whee 0 < x, ρ >0 and ν, >0 (.) They found (ibid) the th moment of the above distibution as Othewise E(X ) + Γ ( ρ ), p > ν Γ( ν) (.3) hence the mean, vaiance and coefficient of vaiation of (.) ae + ( ρ ) Γ mean Γν ( ) (.4) Γ+ ( ) ρ+ v ( Γ( ν) ){ ρ p } Vaaiance ( Γ( ) / Γ (v) (.5)

3 36 Coefficientof va iation { } pv+ pv+ Γ(v) Γ( ) Γ ( ) p p Γ( ) (pv+ ) p (.6) Gupta (984) futhe genealized the genealized Poisson (GP) of Consul and Jain (973a) by adding a thid paamete ω>0, giving a genealized fom as (GGP) (.7) x x θ( θ + θx) ( ωexp( θω)) f(x; θ, θ, ω ) x!exp( θω ) x 0,,,3,... ωθ > 0, ωθ < MIXTURE Let T {F} be a family of cumulative distibution functions and µ a measue defined on Boel field of subsets of T with µ(t), then H( i) F( i ) µ (F) T (.) is again a distibution function called a µ-mixtue of T.. Mixtue of (GGP) and (GG) Fom (.7) the (GGP) is f(x; θ, θ, ω ) x θ( θ + θx) ( ωexp( θω)) x!exp( θω) x 0,,,3,... ωθ > 0, ωθ < x Hee we shall keep θ and ω fixed and assume that θ has the GG distibution having the mixing density function f(x; ν,,ρ) ρθ - θ exp{- ρ} Γ( ν ) Whee 0 < x, ρ >0 and ν, >0 Hee, ν, p ae known constants. The mixtue of the density function f(x; θ, θ, ω) is fo x0

4 37 f(x; ν,,ρ) 0 - θ exp( θ ω) ρ θ exp{- ρ} Γ( ν) dθ (.3) and fo x 0 we have p Γ(pv) ( ω + p) Γ(v) pv x x - θ θ( θ + θx) ( ωexp( θω)) ρ θ exp{- ρ} 0 x! Γ( ν)exp( θω) f(x; ν,,ρ, θ, ω) dθ x- x - θ x- x ρ ( ωexp( θω)) θ exp{- ρ} ( x) θ θ 0 dθ x! Γ( ν)exp( θ ω) 0 x - x- ρ+ω pv+ x 0 ρ ( ωexp( θ ω)) θ x- ( θ x) Γ( + x ) x! Γ( ν) ( ) ( θx) ( )( ) x x - x- x! x x x! Γ( ν) ω 0 X ( ) ( ) ( ) +ω +ω ρ ( ωexp( θ ω)) θ x We note that the expession in culed backets within squae backets in the above elation is negative binomial pobability P(Yx-) with ( ) ρ +ω and q ω ( +ω ). This should not supise us. Hee we ecall that ou GGP in (.) educes to Poisson distibution with paamete θ by putting θ 0 ω in thee. In this case, the above paamete is, as expected, the negative binomial distibution with ( ) ρ + and q ( + ). Thus the mixtue is

5 38 ργ ( ) ( ) ω + ρ, x 0 Γν ( ) ( )( ) x x ( θx) h(x; νρϑ,,,, ω), x 0 x - x- x! ρ ( ω exp( θ ω )) θ x x x! Γ( ν) ω 0 X ( ) ( ) ( ) +ω +ω.5 Lemma llet µ f(k, θ ) and µ h(k) denote the k th aw (o cental moments, espectively, of the fequency function f(x; θ) and h(x,δ). Then µ (k) µ (k, θ)g( θ, δ)dθ h θ f.6 Fom (.6) it I easily seen that the mean, vaiance, the thid and fouth cental moments of the mixtue ae.7 mean ω ρ Γ( + ) ρ ( ωθ ) Γ( ν) + ( ).8 va iance ω ρ Γ( + ) ρ 3 ( ωθ) Γ( ν) + ( ).9 ω ρ ( + θ ) Γ( + ) ρ µ ( ) Γ( ν) 3 5 ωθ + ( ) ω ργ+ ( ) ρ ωp Γ+ ( )(+ 8ωθ + 6 θ)( ρ) µ + + ( ) + ( ) ( ωθ) Γ( ν) ( ωθ) Γ( ν).0 Thus. ( + ) ( + ωθ )( Γ( ν)) ρ { ρω( ωθ ) Γ+ ( )} β

6 39. 3 ρ ωγ( ν) Γ( + ) ρ ( + 8ωθ + 6 ωθ ) Γ( ν) ρ ( + ) + β + Γ ( + ) ρ ω ( ωθ ) ρ Γ ( + ) We obseve that, as in the case of GGP distibution the vaiance of the mixtue is geate than, equal to o less than the mean accoding as the poduct ωθ is positive, zeo o negative and ωθ detemines how lage o small the mean and vaiance would be. We also obseve that the expession fo β indicates that the skewness of the mixtue is positive if ωθ > -0.5 and negative when - < ωθ < -0.5 just as in GGP. The kutosis of the mixtue is in the limit equal to that of nomal distibution as v tends to infinity. We also obseve that GGP is a special case of the mixtue when p equals.