Case-deletion importance sampling estimators: Central limit theorems and related results

TL;DR

This paper develops theoretical results to evaluate the reliability of importance sampling estimators in Bayesian case-deletion analysis, focusing on conditions for their asymptotic normality and practical assessment.

Contribution

It provides simple analytical criteria to verify when case-deleted importance sampling estimators satisfy a central limit theorem in Bayesian models.

Findings

Conditions for CLT satisfaction are easily verified.

Theoretical tools for assessing estimator dependability.

Illustrative examples demonstrate practical application.

Abstract

Case-deleted analysis is a popular method for evaluating the influence of a subset of cases on inference. The use of Monte Carlo estimation strategies in complicated Bayesian settings leads naturally to the use of importance sampling techniques to assess the divergence between full-data and case-deleted posteriors and to provide estimates under the case-deleted posteriors. However, the dependability of the importance sampling estimators depends critically on the variability of the case-deleted weights. We provide theoretical results concerning the assessment of the dependability of case-deleted importance sampling estimators in several Bayesian models. In particular, these results allow us to establish whether or not the estimators satisfy a central limit theorem. Because the conditions we derive are of a simple analytical nature, the assessment of the dependability of the estimators can be verified routinely before estimation is performed. We illustrate the use of the results in several examples.