Estimation and influence diagnostics for zero-inflated hyper-Poisson regression model: full Bayesian analysis

Abstract

The purpose of this paper is to develop a Bayesian analysis for the zeroinflated hyper-Poisson model. Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the model and the Bayes estimators are compared by simulation with the maximum-likelihood estimators. Regression modeling and model selection are also discussed and case deletion influence diagnostics are developed for the joint posterior distribution based on the functional Bregman divergence, which includes ψ-divergence and several others, divergence measures, such as the Itakura-Saito, Kullback-Leibler, and χ 2 divergence measures. Performance of our approach is illustrated in artificial, real apple cultivation experiment data, related to apple cultivation.

A new class of regression model for a bounded response with application in the study of the incidence rate of colorectal cancer

“…Let ω(1),…,ω(Q) be a sample of size Q from π(ω|scriptD). Then, a Monte Carlo estimate of K(P,P(-i)) is given by In particular, from equation (17), a Monte Carlo estimate of K–L divergence DK-L(P,P(-i)) is obtained as Values of the Dζ(P,P(-i)) can be interpreted as the ζ -divergence of the effect of deleting of i -th case from the full data on the joint posterior distribution of ω. As pointed out by Peng and Dey, 34 Weiss, 35 and Cancho et al., 17 it may be difficult for a practitioner to judge the cutoff point of the divergence measure to determine whether a small subset of observations from the full data is influential or not. Thus, we will use the calibration proposal given by Peng and Dey…”

“…In this context, although not required for development, we assume that the prior distribution of α , β and τ is independent, that is, π(α,β,τ)=π(α)π(β)π(τ), with α∼π(α),β∼Np(0,Σβ) and γ∼Nq(0,Σγ), where Nr(0,Σ0) denoting the r -variate normal distribution with mean vector 0 and covariance matrix Σ0. Following some authors, see for example Cancho et al., 17 here all the hyper-parameters are specified in order to express weak-informative priors. The hyper-parameter associated with the variance of …”

“…In this context, although not required for development, we assume that the prior distribution of , b and s is independent, that is, ð, b, sÞ ¼ ðÞðbÞðsÞ, with $ ðÞ, b $ N p ð0, AE b Þ and $ N q ð0, AE Þ, where N r ð0, AE 0 Þ denoting the r-variate normal distribution with mean vector 0 and covariance matrix AE 0 . Following some authors, see for example Cancho et al, 17 here all the hyper-parameters are specified in order to express weak-informative priors. The hyper-parameter associated with the variance of b and was fixed in AE b ¼ 100I p and AE ¼ 100I q with I l is an identity matrix of dimension l. More specific details are given in the simulation study and in the application.…”

“…In particular, from equation (17), a Monte Carlo estimate of K-L divergence D KÀL ðP, P ðÀiÞ Þ is obtained as…”

“…Values of the D ðP, P ðÀiÞ Þ can be interpreted as the -divergence of the effect of deleting of i-th case from the full data on the joint posterior distribution of x: As pointed out by Peng and Dey, 34 Weiss, 35 and Cancho et al, 17 it may be difficult for a practitioner to judge the cutoff point of the divergence measure to determine whether a small subset of observations from the full data is influential or not. Thus, we will use the calibration proposal given by Peng and Dey 34 and Weiss.…”