Bayesian Estimators in Uncertain Nested Error Regression Models

TL;DR

This paper introduces a Bayesian nested error regression model with uncertain random effects, using mixture distributions and objective Bayesian inference, validated through simulation and empirical studies.

Contribution

It proposes a novel Bayesian model with uncertain random effects and develops Gibbs sampling for inference, enhancing small area estimation methods.

Findings

Posterior distribution is proper with finite variances.

Gibbs sampling efficiently generates posterior samples.

Model outperforms conventional methods in simulations and empirical tests.

Abstract

Nested error regression models are useful tools for analysis of grouped data, especially in the case of small area estimation. This paper suggests a nested error regression model using uncertain random effects in which the random effect in each area is expressed as a mixture of a normal distribution and a positive mass at $0$. For estimation of the model parameters and prediction of the random effects, an objective Bayesian inference is proposed by setting non-informative prior distributions on the model parameters. Under mild sufficient conditions, it is shown that the posterior distribution is proper and the posterior variances are finite, confirming the validity of posterior inference. To generate samples from the posterior distribution, we provide the Gibbs sampling method with familiar forms for all the full conditional distributions. This paper also addresses the problem of predicting finite population means, and a sampling-based method is suggested to tackle this issue. Finally, the proposed model is compared with the conventional nested error regression model through simulation and empirical studies.