ON ZERO INFLATED GENERALIZED POISSON-SUJATHA DISTRIBUTION AND ITS APPLICATIONS (original) (raw)
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Excessive zeros in multivariate count data are often encountered in practice. Since the Poisson distribution only possesses the property of equi-dispersion, the existing Type I multivariate zero-inflated Poisson distribution (Liu and Tian, 2015, CSDA) [15] cannot be used to model multivariate zero-inflated count data with over-dispersion or under-dispersion. In this paper, we extend the univariate zero-inflated generalized Poisson (ZIGP) distribution to Type I multivariate ZIGP distribution via stochastic representation aiming to model positively correlated multivariate zero-inflated count data with over-dispersion or underdispersion. Its distributional theories and associated properties are derived. Due to the complexity of the ZIGP model, we provide four useful algorithms (a very fast Fisher-scoring algorithm, an expectation/conditional-maximization algorithm, a simple EM algorithm and an explicit majorizationminimization algorithm) for finding maximum likelihood estimates of parameters of interest and develop efficient statistical inference methods for the proposed model. Simulation studies for investigating the accuracy of point estimates and confidence interval estimates and comparing the likelihood ratio test with the score test are conducted. Under both AIC and BIC, our analyses of the two data sets show that Type I multivariate ZIGP model is superior over Type I multivariate zero-inflated Poisson model.
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To model count data with excess zeros and excess ones, in their unpublished manuscript, Melkersson and Olsson (1999) extended the zero-inflated Poisson distribution to a zero-and-one-inflated Poisson (ZOIP) distribution. However, the distributional theory and corresponding properties of the ZOIP have not yet been explored, and likelihoodbased inference methods for parameters of interest were not well developed. In this paper, we extensively study the ZOIP distribution by first constructing five equivalent stochastic representations for the ZOIP random variable and then deriving other important distributional properties. Maximum likelihood estimates of parameters are obtained by both the Fisher scoring and expectation-maximization algorithms. Bootstrap confidence intervals for parameters of interest and testing hypotheses under large sample sizes are provided. Simulations studies are performed and five real data sets are used to illustrate the proposed methods.
Statistics, Optimization & Information Computing, 2015
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 λ 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 λ share a common parameter of zero inflation while the two marginal distributions of Type II bivariate ZIGP λ 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.
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In this paper, a zero-truncated Poisson-Amarendra distribution (ZTPAD), a zero-truncation of Poisson-Amarendra distribution (PAD) of Shanker (2016 b) has been introduced and investigated. A general expression for the rth factorial moment about origin has been obtained and thus the first four moments about origin and the central moments have been given. The expressions for coefficient of variation, skewness, kurtosis, and the index of dispersion of the distribution have been presented and their graphs for varying values of parameter have been given. The condition under which the ZTPAD is over-dispersed, equi-dispersed and under-dispersed has been compared with that of zero-truncated Poisson-Lindley distribution (ZTPLD) and zero-truncated Poisson-Sujatha distribution (ZTPSD). The method of maximum likelihood estimation and the method of moments have been discussed for estimating its parameter. Application of ZTPAD to a real data set has been given and its goodness of fit has been comp...
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A new mixed Poisson model is proposed as a better alternative for modelling count data in the presence of overdispersion and/or heavy-tail. The mathematical properties of the model were derived. The maximum likelihood estimation method is employed to estimate the model’s parameters and its applications to the three real data sets discussed. The model is used to model sets of frequencies that have been used in different literature on the subject. The results of the new model were compared with Poisson, Negative Binomial and Generalized Poisson-Sujatha distributions (POD, NBD and GPSD, respectively). The parameter estimates expected frequencies and the goodness-of-fit statistics under each model are computed using R software. The results show that the proposed PSD fits better than POD, NBD and GPSD for all the data sets considered. Hence, PSD is a better alternative provided to model count data exhibiting overdispersion property.