Impact of existence and nonexistence of pivot on the coverage of empirical best linear prediction intervals for small areas

TL;DR

This paper investigates the coverage accuracy of empirical best linear prediction intervals for small areas, revealing the importance of the existence of a pivot and proposing a double bootstrap correction to improve coverage.

Contribution

It analytically demonstrates the impact of pivot existence on coverage error and introduces a double bootstrap method to correct coverage issues in small area prediction intervals.

Findings

Coverage error is of order O(m^{-3/2}) with a pivot, but degrades without it.

Existing bootstrap methods may overcover due to positive bias in coverage.

Proposed double bootstrap improves coverage accuracy in simulations.

Abstract

We advance the theory of parametric bootstrap in constructing highly efficient empirical best (EB) prediction intervals of small area means. The coverage error of such a prediction interval is of the order $O(m^{-3/2})$, where $m$ is the number of small areas to be pooled using a linear mixed normal model. In the context of an area level model where the random effects follow a non-normal known distribution except possibly for unknown hyperparameters, we analytically show that the order of coverage error of empirical best linear (EBL) prediction interval remains the same even if we relax the normality of the random effects by the existence of pivot for a suitably standardized random effects when hyperpameters are known. Recognizing the challenge of showing existence of a pivot, we develop a simple moment-based method to claim non-existence of pivot. We show that existing parametric bootstrap EBL prediction interval fails to achieve the desired order of the coverage error, i.e. $O(m^{-3/2})$, in absence of a pivot. We obtain a surprising result that the order $O(m^{-1})$ term is always positive under certain conditions indicating possible overcoverage of the existing parametric bootstrap EBL prediction interval. In general, we analytically show for the first time that the coverage problem can be corrected by adopting a suitably devised double parametric bootstrap. Our Monte Carlo simulations show that our proposed single bootstrap method performs reasonably well when compared to rival methods.