Corrected Empirical Bayes Confidence Region in a Multivariate Fay-Herriot Model

TL;DR

This paper develops a second-order correct confidence region for small area means in a multivariate Fay-Herriot model with unknown covariance, using an estimator that is positive-definite and consistent.

Contribution

It introduces a new confidence region based on Mahalanobis distance for multivariate small area estimation with unknown covariance matrices.

Findings

The confidence region is second-order correct.

The proposed estimator of the covariance matrix is positive-definite and consistent.

Simulation studies demonstrate the effectiveness of the method.

Abstract

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) in the linear mixed model is useful because it gives a stable estimate for a mean of a smallarea. For measuring uncertainty of EBLUP, much of research is focused on second-orderunbiased estimation of mean squared prediction errors in the univariate case. In this paper, we consider the multivariate Fay-Herriot model where the covariance matrix of random effects is fully unknown, and obtain a confidence reagion of the small area mean that is based on the Mahalanobis distance centered around EBLUP and is second order correct. A positive-definite, consistent and second-order unbiased estimator of the covariance matrix of the random effects is also suggested. The performance is investigated through simulation study.