Discrete Support Set Selection for Gamma Prior Density Estimation in Measurement Error Model using Empirical Bayes Deconvolution

Fevi  Novkaniza; Khairil  Anwar Notodiputro; I Wayan  Mangku; Kusman  Sadik

Authors

Fevi Novkaniza Department of Statistics, Faculty of Mathematics and Natural Sciences, IPB University, Bogor, West Java, Indonesia
Khairil Anwar Notodiputro Department of Statistics, Faculty of Mathematics and Natural Sciences, IPB University, Bogor, West Java, Indonesia
I Wayan Mangku Department of Mathematics, Faculty of Mathematics and Natural Sciences, IPB University, Bogor, West Java, Indonesia
Kusman Sadik Department of Statistics, Faculty of Mathematics and Natural Sciences, IPB University, Bogor, West Java, Indonesia

Keywords:

Bias, conjugate, hyperparameter, loglikelihood, Poisson

Abstract

This paper discusses the empirical Bayes deconvolution (EBD) method in estimating gamma’s prior density for count data when the true or unobserved random variable is subject to measurement error. The observed random variable $gif.latex?W$ is related to the unobserved random variable by an additive measurement error model. The count data $gif.latex?W_{1},&space;W_{2},&space;...,W_{n}$ are assumed to follow a Poisson distribution as realizations from an unknown prior density $gif.latex?g(x).$ Then the EBD method is applied to estimate $gif.latex?g(x)$ for every discretization point in the discrete support set of $gif.latex?X.$ The effect of selecting discrete support set for estimating gamma’s prior density based on the EBD method is illustrated by using simulation. It is shown that by selecting discretization set for Poisson data and gamma density as a prior distribution, the larger domain, and more points in discrete support set, the smaller value of bias, and standard deviation for gamma prior density estimate. Finally, assuming that the number of high school student dropout follows Poisson distribution, the EBD method is applied to estimate the prior probability distribution for high school student dropout data in 9 cities and 18 districts in West Java province.