Recursive Relation for Zero Inflated Poisson Mixture Distributions

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1 Applied Mathematical Sciences, Vol., 7, no. 8, HIKARI Ltd, Recursive Relation for Zero Inflated Poisson Mixture Distributions Cynthia L. Anyango, Edgar Otumba Department of Mathematics Statistics Maseno University, P.O. Box 333 Maseno, Kenya John M. Kihoro The Cooperative University College of Kenya School of Computing elearning P.O. Box 484 Karen, Kenya Copyright c 7 Cynthia L. Anyango, Edgar Otumba John M. Kihoro. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract The paper extends the work of Sarguta who derived recursive relations for univariate distributions by considering the ZIP continuous mixtures. The paper gives a recursive formular which can be used to evaluate the mixed distributions which can be used when the probability distribution functions cannot be evaluated explicitly. Integration by parts is often employed when deriving the recursive formulas. From section two up to section seven, we derived the recursive formulas for ZIP mixture distributions using Rectangular, Exponential, Gamma with two parameters, Poisson- Beta Inverted - Beta as mixing distributions. Keywords: ZIP, recursive, inflated model, prior distributions integration Introduction A zero-inflated model is a statistical model based on a zero-inflated probability distribution. Gardner et.al in their paper, [9], suggested that using an

2 85 Cynthia L. Anyango, Edgar Otumba John M. Kihoro inflation technique was adequate if the intention is to estimate the effect of the covariates. In April, 9, the University of Carlos the third team in their [], assessed the impacts of the fertility decisions of mothers on infant mortality. They used a Poisson regression to model the number of children. They fitted an inflated zero s model with negative binomial to the fertility decisions so as to eliminate the problem of overdispersion of the Poisson model. A main difficulty with the use of Mixed Poisson distribution is that, with the exception of a few mixing distributions, their probability mass function f(x is difficult to evaluate [8]. One way of circumventing this problem is to express the mixed distributions in terms of recursive relations. A number of methods for deriving such recursive relations have been developed, starting with the works of [5], [4], [], [3], etc. Integration by parts does not require assumptions given by [3] or by []. Rectangular mixing distribution Therefore, the mixed distribution becomes b P rob(y k [ρ + ( a ρe λ ], k; b a [ e λ λ k ], k,,.... k! b a ρ + b a [ b e λ a e λ ], k; [ b k!(b a e λ λ k a e λ λ k ], k,,.... ρ + [ b a e a e b], k; [Γ k!(b a b(k + Γ a (k + ], k,,.... Let us consider the following function, using integration by parts b e λ λ k b e λ λ k λ k e λ b + k b e λ λ k ( [b k e b ] + kγ b k ( Hence Γ b (k + b k e b + kγ b k b k e b kb k e b + k(k Γ b (k b k e b kb k e b + k(k [ b k e b + (k Γ b (k ] e b [b k + kb k + k(k b k + k(k (k b k 3 + +k(k (k (k 3 [k (k ]b k k ]

3 Recursive relation for zero inflated Poisson mixture distributions 85 Therefore k!(b a Γ b(k e b (b a Similarly k!(b a Γ a(k e a (b a a k e b b k P r(y k (b a (e a k! [ b k k! + bk (k! + bk (k! + + b! + ]! [ a k k! + ak (k! + ak (k! + + a! + ]! +( e a a k e b b k (k! + + e a a e b b +(e a e b }! P r(y k + a k+ e b b k+ (b a [(e a + ( e a a k e b b k + + (k! k! ( e a a e b b + (e a e b ] a k+ e b b k+ (b a e a } + P r(y k (k +! The recursive formula becomes ρ + b a P rob(y k + [e a e b ], k; e a a k+ e b b k+ } + P r(y k, k,,.... (3 (b a (k+! with P r(y 3 Poisson-Inverse Gaussian Distribution If the Inverse Gaussian mixing distribution is given by φ g(λ ( πλ φ(λ µ 3 exp } λ >, µ >, φ > µ λ then the recursive formula for Zero Inflated Poisson-Inverse Gaussian distribution becomes P r(y k [ρ + e λ ]( φ πλ 3 exp φ(λ µ }, k; µ λ [ e λ λ k ]( φ k! πλ 3 exp φ(λ µ }, k,,.... µ λ ρ + ( φ e φ µ λ 3 e λ(+ φ, k; π ( φ e φ µ λ k 3 e λ(+ φ, k,,.... k! π Let I k λ k 3 e λ(+ φ

4 85 Cynthia L. Anyango, Edgar Otumba John M. Kihoro Using integration by parts, let then This implies that du λ k+ 3 I k k + 3 ( + φ µ u e λ(+ φ dv λ k 3 [ ( + φ µ + φ ] e λ(+ φ λ v λk+ 3 k + 3 [ ( + φ (k + 3 φ (k + 3 ( + φ µ µ + φ λ ] e λ(+ φ λ k+ 3 e λ(+ φ λ k 3 e λ(+ φ (k + 3 I k+ φ (k + 3 I k This implies that ( k!p r(y k + φ (k +! φ(k! µ (k + 3 r(y k+ P (k + 3 r(y k P kp r(y k ( + φ k(k + φ µ (k + 3 r(y k + P r(y k P k Therefore, the recursive formula is [ with ( + φ k(k + µ (k + 3 r(y k + P [ kp r(y k + P r(y ] φ ] P r(y k k (4

5 Recursive relation for zero inflated Poisson mixture distributions Poisson-Exponential with One parameter If the distribution for the exponential with one parameter is given by g(λ µ e µλ λ >, µ >, then the recursive formula for the Zero Inflated Poisson-Exponential with one parameter distribution becomes ρ + µ P r(y k e (+µλ, k; µ e (+µλ λ k k,,.... k! Let I k Using integration by parts,, then Then It follows that Therefore I k The recursive formular k!p r(y k u e (+µλ dv λ k du ( + µe (+µλ k + ( + µ k + I k+ (k +!P r(y k + µ P r(y k + P r(y k + µ e (+µλ λ k ( + µe (+µλ λ k+ I k+ ( k + I k + µ ( k + + µ ( + µ ( k + k! P r(y k + µ µ k! P r(y k (k +! P r(y k ρ + µ (, (+µ k; P r(y k, k,,.... +µ (5

6 854 Cynthia L. Anyango, Edgar Otumba John M. Kihoro 5 Poisson-Gamma with Two parameters If the pdf of a Gamma distribution with two parameters is given by g(λ α Γα e λ λ α, λ >. α >, > then the recursive formula for Zero Inflated Poisson-Gamma with two parameters becomes P r(y k ρ + α k! α Γ(α Γ(α e (+λ λ α, k; e (+λ λ α +k, k,,.... Now Using integration by parts k! Γα P (Y k I α k e (+λ λ α +k I k ( + + α + k e (+λ λ α +k I k+ α + k Then It follows that I k+ ( α + k I k + (k +! Γα (α + k k! P r(y k + α ( + Γα P r(y k α Thus ( P r(y k + α + k ( + (k + P r(y k The recursive formula becomes ( α ρ + +, k; P r(y k + ( α+k (k+(+ P r(y k, k,,.... (6 6 Mixing with Poisson - Beta distribution The Poisson - Beta distribution is g(λ λα ( λ, λ B(α,

7 Recursive relation for zero inflated Poisson mixture distributions 855 The recursive formula is derived as follows; P r(y k ρ + e λ λ α ( λ k! e λ λ k λ α ( λ ρ + B(α, k!b(α, B(α,, k;, k,,.... B(α, e λ λ α ( λ, k; e λ λ α+k ( λ, k,,.... Now, I k (α, k!b(α, P r(y k e λ λ α+k ( λ Let Therefore Therefore u e λ λ α+k dv ( λ du e λ λ α+k + (α + k e λ λ α+k ( λ v I k (α, Thus e λ λ α+k ( λ + (α + k e λ λ α+k ( λ e λ λ α+k ( λ( λ + (α + k e λ λ α+k ( λ( λ e λ λ α+k ( λ e λ λ α+k ( λ } + α + k I k(α, I k+ (α, } + e λ λ α+k ( λ e λ λ α+k ( λ } (α + k I k (α, I k (α, } I k+ (α, I k (α, + I k (α, + (α + k I k (α, (α + k I k (α, (α + + ki k (α, (α + k I k (α,

8 856 Cynthia L. Anyango, Edgar Otumba John M. Kihoro This implies that (k+!p r(y k+ (α++kk!p r(y k (α+k (k!p r(y k (7 Hence the recursive formular is k(k+p r(y k+ (α++kkp r(y k (α+k P r(y k (8 with P r(y 7 Mixing with Inverted - Beta distribution The mixing distribution is g(λ The mixed distribution is ρ + P r(y k λ α, λ >, α >, > B(α, ( + λ α+ k! λ λα e, B(α, (+λ α+ e λ λ k λα k; B(α, (+λ α+, k,,.... Let then I k (α + (α + (α + k (α + (α + k!b(α, e λ λ α+k P r(y k ( + λ I α+ x u e λ λ α+k dv ( + λ α+ du e λ λ α+k + e λ (α + k λ α+k ( + λ (α+ v (α + ( + λ (α+ e λ (α + k λ α+k e λ λ α+k } e λ (α + k λ α+k ( + λ ( + λ( + λ (α+ e λ λ α+k ( + λ (α+ + e λ λ α+k + ( + λ (α+ (α + k (α + I k + I k } (α + I k + I k+ } e λ λ α+k } ( + λ (α+ e λ λ α+k } ( + λ (α+ e λ λ α+k ( + λ } ( + λ( + λ (α+

9 Recursive relation for zero inflated Poisson mixture distributions 857 Therefore which implies that (α + I k (α + k (I k + I k (I k + I k+ Thus I k+ (α + k I k + (k I k (k +!B(α, P r(y k + + (k k!b(α, P r(y k (α + k (k!b(α, P r(y k (9 which when simplified gives the recursive formular a Zero Inflated Poisson- Inverted Beta distribution as k(k + P r(y k + k(k P r(y k + (α + k P r(y k, with P r(y 8 Conclusion From the above continuous prior distributions, it can be clearly seen that the recursive relations can be derived for numerous mixture distributions. This is made possible by the fact that there are no restrictions imposed during integration. It is known that the distributions do not exist when the variable k <. We restricted ourselves to continuous mixing distribution even though, discrete or countable mixtures where we have discrete prior distributions could be of interest to a researcher, thus, research can be carried out on this. Acknowledgements. Special thanks to Prof. J. A. M. Ottieno for his valuable comments input towards this paper. References [] B. Sundt, On some Extensions of Panjer s dass of distributions, ASTIN Bulletin, (99, [] D. Karlis E. Xekalaki, Mixed Poisson Distribution, International Statistical Review, 73 (5,

10 858 Cynthia L. Anyango, Edgar Otumba John M. Kihoro [3] G. E Willmot, On recursive evaluation of mixed poisson probabilities related quantities, Scinavian Actuarial Journal, (993, [4] H, H Panjer, Recursive evaluation of a family of compound distributions, ASTIN Bulletin, (98, [5] L. Katz, Unified treatment of a broad class of discrete probability distributions, Classical Contagious Discrete Distributions, Pergamon Press, Oxford, 965. [6] M. Greenwood G. Yule, An inquiry into the nature of frequency distribution representative of multiple happenings with particular reference to the occurence of multiple attacks of disease or repeated accidents, Journal of Royal Statistical Society, 83 (9, [7] M. Perakis E. Xekalaki, A process capability index for discrete processes, Journal of Statistical Computation Simulation, 75 (5, no. 3, [8] P. Albrecht, Laplace Transforms, Mellin Transforms Mixed Poisson Processes, Scinavian Atuarial Journal, 984 (984, [9] W. Gardner, E. P Mulvey E. C Shaw, Regression analysis of counts rates: Poisson, overdispersed Poisson negative binomial model, Psychological Bulletin, 8 (995, [] University of Madrid Carlos III, Assessing the Impacts of Infant Mortality Upon Fertility Decisions of Mothers in India, AENORM 63, 9. Received: March 3, 7; Published: March 3, 7