Abstract
To model correlated bivariate count data with extra zero observations, this paper proposes two new bivariate zero-inflated generalized Poisson (ZIGP) distributions by incorporating a multiplicative factor (or dependency parameter) λ, named as Type I and Type II bivariate ZIGP distributions, respectively. The proposed distributions possess a flexible correlation structure and can be used to fit either positively or negatively correlated and either over- or under-dispersed count data, comparing to the existing models that can only fit positively correlated count data with over-dispersion. The two marginal distributions of Type I bivariate ZIGP share a common parameter of zero inflation while the two marginal distributions of Type II bivariate ZIGP have their own parameters of zero inflation, resulting in a much wider range of applications. The important distributional properties are explored and some useful statistical inference methods including maximum likelihood estimations of parameters, standard errors estimation, bootstrap confidence intervals and related testing hypotheses are developed for the two distributions. A real data are thoroughly analyzed by using the proposed distributions and statistical methods. Several simulation studies are conducted to evaluate the performance of the proposed methods.
Highlights
As the simplest distribution for modeling count data, the Poisson distribution possesses an exceptional property of equi-dispersion, i.e., its mean and variance are identical
When there is a larger frequency of (0,0) observations in the bivariate count data, the issue of the over-dispersion may arise. To model such extra zero points in the data, we propose a new bivariate zero-inflated generalized Poisson (ZIGP) distribution indexed by a multiplicative factor λ ∈ R via a mixture of a Bernoulli variable with the bivariate generalized Poisson (GP) distribution (1)
Performance of the likelihood ratio test (LRT) in Type II bivariate ZIGPλ In this subsection, we investigate the performance of the LRT for the testing hypotheses (36) and (39) for the Type II bivariate ZIGPλ distribution
Summary
As the simplest distribution for modeling count data, the Poisson distribution possesses an exceptional property of equi-dispersion, i.e., its mean and variance are identical. To model count data with both over-dispersion and/or under-dispersion, the generalized Poisson (GP) distribution can be used and its probability mass function (pmf) with two parameters is defined by ([7], p.5). The aim of this paper is to develop two new bivariate zero-inflated generalized Poisson (ZIGP) distributions by incorporating a multiplicative factor (or dependency parameter) λ, named as Type I and Type II bivariate. 2. Type I bivariate zero-inflated generalized Poisson distribution with a multiplicative factor. When there is a larger frequency of (0,0) observations in the bivariate count data, the issue of the over-dispersion may arise To model such extra zero points in the data, we propose a new bivariate ZIGP distribution indexed by a multiplicative factor λ ∈ R via a mixture of a Bernoulli variable with the bivariate GP distribution (1).
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