Two new bivariate zero-inflated generalized Poisson distributions with a flexible correlation structure

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 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.

Evaluating capability of a bivariate zero-inflated poisson process

“…Then, using the joint distribution, they developed a bivariate control chart which can be used for monitoring correlated rare events in a BZIP process. Liu and Tian (2015) proposed a Type I multivariate ZIP distribution to model correlated multivariate count data with extra zeroes. The bivariate ZIP distribution having only three parameters, derived from the proposed MZIP distribution, is much simpler in comparison with Li et al (1999) proposed BZIP distribution and copula-based BZIP distribution proposed by Fatahi et al (2012).…”

“…Fatahi et al (2012) applied the copula function approach to develop the joint distribution of two correlated ZIP distributions. Liu and Tian (2015) proposed a Type I multivariate ZIP distribution to model correlated multivariate count data with extra zeroes. They also described methods for obtaining maximum likelihood estimates of the proposed Type I MZIP distribution.…”

“…They also described methods for obtaining maximum likelihood estimates of the proposed Type I MZIP distribution. Zhang et al (2015) proposed two new bivariate zero-inflated generalized Poisson (BZIGP) distribution with a flexible correlation structure by incorporating a multiplicative parameter. Wang (2003) developed a bivariate zero-inflated negative binomial regression model for bivariate count data with excess zeroes.…”

“…The article is organized as follows: in Section 2, we describe two different sets of BZIP distributions derived from Li et al (1999) MZIP distribution andLiu andTian (2015) Type I MZIP distribution for modelling bivariate zero-inflated count data. In section 3, we propose our approach for computing process capability index ‫ܥ‬ ௨ ூ for the bivariate zero-inflated Poisson process.…”

Type I multivariate zero‐inflated COM–Poisson regression model

“…Zhang et al. (2015) develop two new bivariate zero‐inflated generalized Poisson distributions (BZIGP) to model correlated bivariate count data with extra zero observations. These distributions also use the Type I MZIP distribution structure proposed by Liu and Tian (2015).…”