On Measuring the Variability of Small Area Estimators in a Multivariate Fay-Herriot Model

TL;DR

This paper develops methods to measure the variability of small area estimators in a multivariate Fay-Herriot model with unknown covariance matrices, providing second-order approximations and unbiased estimators.

Contribution

It introduces a second-order approximation of the mean squared error matrix and an unbiased estimator for small area estimators in a multivariate Fay-Herriot model with unknown covariance.

Findings

The proposed estimators perform well in numerical studies.

The second-order approximation accurately captures the variability.

Empirical results validate the theoretical developments.

Abstract

This paper is concerned with the small area estimation in the multivariate Fay-Herriot model where covariance matrix of random effects are fully unknown. The covariance matrix is estimated by a Prasad-Rao type consistent estimator, and the empirical best linear un- biased predictor (EBLUP) of a vector of small area characteristics is provided. When the EBLUP is measured in terms of a mean squared error matrix (MSEM), a second-order approximation of MSEM of the EBLUP and a second-order unbiased estimator of the MSEM is derived analytically in closed forms. The performance is investigated through numerical and empirical studies.