Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression

Abstract

Multivariate Poisson regression is used in order to model two or more count response variables. The Poisson regression has a strict assumption, that is the mean and the variance of response variables are equal (equidispersion). Practically, the variance can be larger than the mean (overdispersion). Thus, a suitable method for modelling these kind of data needs to be developed. One alternative model to overcome the overdispersion issue in the multi-count response variables is the Multivariate Poisson Inverse Gaussian Regression (MPIGR) model, which is extended with an exposure variable. Additionally, a modification of Bessel function that contain factorial functions is proposed in this work to make it computable. The objective of this study is to develop the parameter estimation and hypothesis testing of the MPIGR model. The parameter estimation uses the Maximum Likelihood Estimation (MLE) method, followed by the Newton–Raphson iteration. The hypothesis testing is constructed using the Maximum Likelihood Ratio Test (MLRT) method. The MPIGR model that has been developed is then applied to regress three response variables, i.e., the number of infant mortality, the number of under-five children mortality, and the number of maternal mortality on eight predictors. The unit observation is the cities and municipalities in Java Island, Indonesia. The empirical results show that three response variables that are previously mentioned are significantly affected by all predictors.

Multivariate mixed Poisson Generalized Inverse Gaussian INAR(1) regression

“…5, we will discuss in detail the distribution function f ( , t) for some special cases. Finally, it should be noted that several articles discuss multivariate versions of MPGIG distribution and/or the MPIG distribution which is a special case for = −0.5 , see, for instance, Barndorff-Nielsen et al (1992), Ghitany et al (2012) Amalia et al (2017, Mardalena et al (2020), Tzougas and di Cerchiara (2021b) and Mardalena et al (2021). However, this is the first time that the MMPGIG-INAR(1) distribution family of INAR(1) models driven by mixed Poisson regression innovations are considered for modelling time series of count response variables.…”

The Geographically Weighted Multivariate Poisson Inverse Gaussian Regression Model and Its Applications

“…In this study, the PIGR model was chosen because this model performs better when modeling data with high overdispersion. In this study, the PIGR model was developed into a multivariate model with two or more response variables [9,10].…”

“…Therefore, the PIGR model can accommodate greater overdispersion than the NBR model can. The PIGR model has been developed into a multivariate model with two or more response variables [10,17]. Globally, the bivariate Poisson inverse Gaussian regression (BPIGR) model was developed by [9], while the multivariate Poisson inverse Gaussian regression (MPIGR) was developed by [10,18].…”

“…The PIGR model has been developed into a multivariate model with two or more response variables [10,17]. Globally, the bivariate Poisson inverse Gaussian regression (BPIGR) model was developed by [9], while the multivariate Poisson inverse Gaussian regression (MPIGR) was developed by [10,18]. Locally, the Geographically Weighted Poisson Inverse Gaussian Regression (GWPIGR) model was created by [19].…”

A Logit Model for Bivariate Binary Responses

“…On the other hand, the BHHH method can be used as an alternative to the numerical optimization method when the elements of the Hessian matrix are unavailable. Following [29], the maximum likelihood ratio test (MLRT) method was used to test the significance of parameters both simultaneously and partially. The performance of the BBL model was evaluated using an empirical study.…”