Bias Corrected Variance Stabilizing Transformation for Small Area Estimation

TL;DR

This paper introduces bias-corrected empirical Bayes estimators for small area means using variance stabilizing transformations, addressing bias issues in back transformations, with theoretical, simulation, and data analysis validation.

Contribution

It proposes asymptotically unbiased EB estimators for small area means that correct bias in back transformations, enhancing existing variance stabilizing methods.

Findings

Bias correction improves estimator accuracy

Theoretical MSE formulas are derived and validated

Connections to NEF-QVF distributions are established

Abstract

Small area estimation models are typically based on the normality assumption of response variables. More recently, attention has been drawn to the transformation of the original variables to justify the assumption of normality. Variance stabilizing transformation of observation serves the dual purpose of reaching closer to normality, as well as known variance of the transformed variables in contrast to the assumption of known variances of the original variables, the latter needed to avoid non-identifiability. However, the existing literature on the topic ignores a certain bias introduced in the seemingly correct back transformation. The present paper rectifies this deficiency by introducing asymptotically unbiased empirical Bayes (EB) estimators of small area means. Mean squared errors (MSEs) and estimated MSEs of such estimators are provided. The theoretical results were accompanied with simulations and data analysis. A somewhat surprising phenomenon is a finding which connects one of our results to the natural exponential family quadratic variance function (NEF-QVF) family of distributions introduced by Morris (1982,1983).