Zero Inflated Poisson Mixture Distributions

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1 Applied Mathematical Sciences, Vol., 27, no. 7, HIKARI Ltd, Zero Inflated Poisson Miture Distributions Cynthia L. Anyango, Edgar Otumba and John M. Kihoro Department of Mathematics and Statistics Maseno University, P.O. Bo 333 Maseno, Kenya Copyright c 27 Cynthia L. Anyango, Edgar Otumba and John M. Kihoro. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper presents a method of constructing the Zero inflated Poisson continuous miture distributions which have applications in various fields. The distributions can be formed by either direct integration or integration of moments of the underlying distributions. The two methods are presented in sections one and two. In section four, we present the mied distributions. We further proved the identities that resulted when the resultant mied distributions were equated. Keywords: ZIP, continuous mied distributions, moments and eplicit Introduction In applications involving count data, it is common to encounter the frequency of observed zeros significantly higher than predicted by the model based on the standard parametric family of discrete distributions. In such situations, zeroinflated Poisson and zero-inflated negative binomial distribution have been widely used in modeling the data, yet other models may be more appropriate in handling the data with ecess zeros, as shown by 2 The consequences of this is misspecifying the statistical model leading to erroneous conclusions and bringing uncertainty into research and practice. Therefore the problem is to identify by constructing other alternatives, to the models already present in the literature that may be more appropriate for modeling data with ecess

2 84 Cynthia L. Anyango, Edgar Otumba and John M. Kihoro zeros. In their paper,, presented a method on how to construct the mied poisson distributions eplicitly. The paper gives an alternative method where the distributions can be constructed using the r th moment epectation of the underlying distribution. The identities that result from the two methods are also proved. A mied distribution is constructed when two probability distributions are mied. Consider a probability distribution whose parameter is varying and also has a distribution. Then the integral or summation of these two distributions forms a mied probability distribution. In 3, constructed a Negative Binomial distribution by miing a Poisson distribution with its parameter. In this work, the ZIP Zero- Inflated Poisson) distribution was mied with various continuous distributions to construct the ZIP miture distributions. 2 Construction of mied ZIP distribution using moments of the miing distribution 2. General case Suppose gλ) is the miing distribution, then ZIP mied distribution, which can be written in terms of the r th moment of the miing distribution as { ρ ρ) j ρ) j j j!k! ) j j! λ j gλ)dλ, k; λ jk gλ)dλ, k, 2,.... ) Letj, then f) can be written as { ρ ρ) ) EΛ ), k; P ry )! ρ) ) EΛ k ), k, 2,.... where EΛ r ) is the r th moment of the miing distribution. 2) 3 Construction of the ZIP distributions by direct integration A random variable Y follows a zero-inflated Mied Poisson distribution with miing distribution having probability density function g if its probability function is given by; P roby k) { ρ ρ)e λ gλ)dλ, k; ρ) e λ λ k k! gλ)dλ, k, 2,.... 3) If we equate equation 4 and equation 5, we get identities. This is so because the resultant distributions should be identical.

3 Zero inflated Poisson miture distributions 84 4 Special cases 4. Eponential distribution For an eponential distribution, the r th moment is EΛ r ) Let λ r e λ dλ λ r e λ dλ y λ, λ λ, anddλ dy EΛ r ) Thus from equation 2, we have P ry ) r! r { ρ ρ) ρ) ) ) r y y dy e )!!, k; k)!, k k, 2,.... From equation 3 we construct the mied ZIP distribution as follows ρ ρ) ) P ry k) ), ) k; k 5) ρ), k, 2,.... From equation 2 and equation 3, we have the following identities 4). 2. ρ) ρ ρ) ) ) )! ρ ρ)! ) ) k k)! ρ) k Proofs From the first identity,

4 842 Cynthia L. Anyango, Edgar Otumba and John M. Kihoro The second identity )!! ) ) ) Γk ) k! k ) k)!!γk ) 4.2 Gamma distribution Γk ) k! k Γk ) k! k Γk ) k! k k! k! k k ) ) ) ) k k ) ) k) ) k ) ) k ) From equation 5, ρ ρ) β β )α, k; ) P ry k) α k ) α ) k β 6) ρ) k β β, k, 2,.... The mied distribution is a ZINB distribution with parameters α, ρ and From equation 4, The mied distribution becomes β β. P ry k) { ρ ρ) ρ) ) ) β α Γα)! Γα β α Γα, β α k; Γαk), β αk k, 2,.... 7) Identities a) ρ ρ) ) ) β α α Γα ) β ρ ρ)! Γα β α β

5 Zero inflated Poisson miture distributions 843 b) Proof; ρ) ) β α Γα k) α k ρ) Γα β αk k ) β α β) αk ) Γα )! Γα β For the second identity α ) ) ) α α β ) α β ) α β β ) β Γα k) ) Γαβ k Γα k) β k k!γα Γα k) k!γα Γα k) k!γα Γα k) k!γα β k α k k α k k Γα k) )!Γα k) β k α k ) β k ) α k) β k β β ) β α β k ) αk) β) αk ) β β β k ) α β ) αk ) k ) ) β 4.3 Generalized Lindley distribution The mied distribution is { ρ ρ) α P ry k) ρ) k! ) α ) α ) α α Γαk) ) αk Γα) α αk k; k, 2,.... 8)

6 844 Cynthia L. Anyango, Edgar Otumba and John M. Kihoro which is a Zero-Inflated Generalized Poisson Lindley distribution with two parameters. Using the method of moments, ρ ρ) ) α αγα) Γα), k;! )Γα) P ry k) α α Identities a) b) Proof; ρ) ρ ρ) α α α α ρ) ) α )Γα) αγαk) αk Γαk), k, 2,.... αk 9) ) α αγα Γα! )Γα ) α α ρ ρ) α ) ) α ) α ) ρ) k! α α α αγα k) )Γα ) αk Γα k ) αk α αγα k) Γα k ) )Γα ) ) αk ) αk Γα ) )!Γα ) α α α α Γα ) )!Γα Γα ) )!Γα ) α ) ) Γα ) )!Γα ) 2) ) ) α ) ) ) α α ) 3) ) ) α α ) ) α α ) α 4) ) α α ) α) 5) ) α ) α α 6) ) α ) α 7)

7 Zero inflated Poisson miture distributions 845 Second item α Γ α k) ) Γ k α ) 9) ) 8) kα! αk! αγk α) Γ α k) Γk α ) Γ k α ) kα )!Γk α) αk )!Γk α ) 2) αγα k) ) ) k α kα ) Γk α ) ) ) k α αk ) 2) αγα k) ) ) kα k α) Γk α ) ) ) kα k α ) kα αk 22) αγα k) kα αγα k) kα ) kα) Γk α ) αk ) kα Γk α ) αk ) kα) 23) ) kα 24) αγα k) Γk α ) 25) ) kα ) αk 4.4 Lindley distribution The mied distribution by direct integration gives ρ ρ) 2 P ry k) 2k ρ) 2 ) k3 ) 2, k;, k, 2, ) By the method of moments, we have ρ ρ) )! P ry k) ρ) 2 This implies that we have two identities, i.e 2 Γ2) 2 Γk2) k2 Γ), k; Γk), k, 2,.... k 27). ρ ρ) )! 2 ρ ρ) 2 Γ 2) 2 ) 2 Γ ) ) 28)

8 846 Cynthia L. Anyango, Edgar Otumba and John M. Kihoro 2. ρ) ) 2 Γ k 2) k2 Γ k ) k ρ) 2 2 k ) k2 Proofs; )! Γ 2) 2 Γ ) Γ2 2 Γ 2 Γ 2) )!Γ2 ) 2 ) 2 ) ) ) For part ii) Γk 2) k2 Γk 2) k! k2 Γk 2) k! k2 Γk 2) k! k2 k )k! k! k2 2 k ) k2 Γ k 2) Γk ) Γ k ) ) Γk 2) k ) Γk )! 29) k 2 ) ) ) k2 k 2) ) k2) Γk ) k! k ) k2 Γk ) k! k ) ) Γk ) k! k Γk ) k! k ) k) ) k k ) ) ) k k ) 5 Discussion The continuous miture distributions can be formed by direct integration. This applies to most distributions since there are no restrictions imposed during integration. In 4, showed that their is a relationship between the moment of an underlying distribution and the distribution function of the mied distribution. This paper has presented such a relationship by using Eponential, Gamma with two parameters, Lindley and the Generalized Lindley distributions as the underlying distributions being mied with the ZIP distribution. Whether direct integration is used or the method of epectation of moments, the resulting distributions must be identical but not necessarily the same. We went ahead and proved identities that resulted by equating mied distributions formed by direct integration and those formed through r th moment epectation of the ) )

9 Zero inflated Poisson miture distributions 847 miing distributions. Conclusion This work concentrated on purely construction. Since we did not ehaust all the available continuous distributions, therefore more work can be done by considering the distributions which we did not use, by using the methods of construction already used and also other methods can be studied or researched on. Mied Poisson distributions ehibit several interesting properties as given by. These properties include; Identifiability and Shape properties, Infinite divisibility, Posterior Moments, etc. The study of these properties can form a good basis of further research on the mied ZIP distributions constructed in this work. In this study, we restricted ourselves to continuous miing distribution. Discrete or countable mitures where we have discrete prior distributions could be of interest to a researcher, thus, research can be carried out on this. References D. Karlis and E. Xekalaki, Mied Poisson Distribution, International Statistical Review, 73 25), W. Gardner, E. P Mulvey and E. C Shaw, Regression analysis of counts and rates: Poisson, overdispersed Poisson and negative binomial model, Psychological Bulletin, 8 995), M. Greenwood and 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 92), M. Perakis and E. Xekalaki, A process capability inde for discrete processes, Journal of Statistical Computation and Simulation, 75 25), no. 3, Received: March, 27; Published: March 23, 27