Weighted Poisson Distributions for Overdispersion and Underdispersion Situations

Abstract. In the literature, there are two probabilistic models of bivariate Poisson : the model according to Holgate and the model according to Berkhout and Plug. These two models express themselves by their probability mass function. The model of Holgate puts in evidence a strictly positive correlation, which is not always realistic. To remedy this problem, Berkhout and Plug proposed a bivariate Poisson distribution accepting the correlation as well negative, equal to zero, that positive. In this paper, we show that these models are nearly everywhere asymptotically equal. From this survey that the φ-divergence converges toward zero, both models are therefore nearly everywhere equal. Also, the model of Holgate converges toward the one of Berkhout and Plug. Some graphs will be presented for illustrating this comparison.

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“…We note that for all η = 0, var (Z 2 ) − E (Z 2 ) > 0, which implies that Z 2 is overdispersed (Castillo et al (1998) …”

Section: Class Of Property (Ii)mentioning

confidence: 96%

Comparison between two bivariate Poisson distributions through the phi-divergence

Nganga¹,

Bidounga²,

Mizère³

et al. 2017

jas

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“…Castillo and Casany (1998) proposed weighted Poisson distribution by considering the Poisson distribution with parameter λ and the weight function w (k) = (k + α) r . They developed the weighted Poisson distribution with the probability function:…”

Section: Weighted Distributionsmentioning

confidence: 99%

Weighted Distributions: A Brief Review, Perspective and Characterizations

Saghir¹,

Hamedani²,

Tazeem³

et al. 2017

IJSP

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The weighted distributions are widely used in many fields such as medicine, ecology and reliability, to name a few, for the development of proper statistical models. Weighted distributions are milestone for efficient modeling of statistical data and prediction when the standard distributions are not appropriate. A good deal of studies related to the weight distributions have been published in the literature. In this article, a brief review of these distributions is carried out. Implications of the differing weight models for future research as well as some possible strategies are discussed. Finally, characterizations of these distributions based on a simple relationship between two truncated moments are presented.

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“…In the literature, such data can be modelled through some weighted or generalized form of the Poisson distributions [1], but these models do not belong to the exponential family of distributions. Recently, Shmueli et al [7] revived the two-parameter Com-Poisson distribution (CMP), which satisfies the properties of the generalized linear models and can model all forms of dispersion.…”

Section: Introductionmentioning

confidence: 99%

“…The performance of this GQL equation is then assessed on simulation studies and on real-life data applications. Note that there is not yet an algorithm to generate non-stationary AR (1) longitudinal Com-Poisson counts. Thus, we also develop an algorithm to generate such types of counts based on the binomial thinning operation.…”

Section: Introductionmentioning

confidence: 99%

A robust algorithm for estimating regression and dispersion parameters in non-stationary longitudinally correlated Com–Poisson data

Khan¹

2016

LMS J. Comput. Math.

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In recent years, Com-Poisson has emerged as one of the most popular discrete models in the analysis of count data owing to its flexibility in handling different types of dispersion. However, in a stationary longitudinal Com-Poisson count data set-up where the covariates are time independent, estimation of regression and dispersion parameters based on a generalized quasi-likelihood (GQL) approach involves some major computational difficulties particularly in the inversion of the joint covariance matrix. On the other hand, in practical real-life longitudinal studies, time-dependent covariates leading to non-stationary responses are more frequently encountered. This implies that further computational problems will now arise when estimating parameters under non-stationary set-ups. This paper overcomes this problem by approximating the inverse of the ill-conditioned covariance matrix in the GQL approach through a multidimensional conjugate gradient method. The performance of this novel version of the GQL approach is then assessed on simulations of AR(1) stationary and AR(1) non-stationary longitudinal Com-Poisson counts and on real-life epileptic seizure counts. However, there is not yet an algorithm to generate non-stationary longitudinal Com-Poisson counts nor a GQL algorithm to estimate the parameters under non-stationary set-ups. Thus, the paper also provides a framework to generate non-stationary AR(1) Com-Poisson counts along with the construction of a GQL equation under non-stationary set-ups.

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Weighted Poisson Distributions for Overdispersion and Underdispersion Situations

Cited by 68 publications

References 14 publications

Comparison between two bivariate Poisson distributions through the phi-divergence

Comparison between two bivariate Poisson distributions through the phi-divergence

Weighted Distributions: A Brief Review, Perspective and Characterizations

A robust algorithm for estimating regression and dispersion parameters in non-stationary longitudinally correlated Com–Poisson data

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