Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models

TL;DR

This paper introduces a parametric bootstrap method to accurately estimate the distribution of EBLUP in linear mixed models, improving prediction interval accuracy especially in small area estimation.

Contribution

It proposes a novel bootstrap approach for distribution estimation of EBLUP, addressing limitations of traditional MSPE-based intervals in linear mixed models.

Findings

Bootstrap histogram closely approximates true EBLUP distribution

Method achieves higher accuracy with $O(d^3n^{-3/2})$ error rate

Simulation shows superior performance over existing interval methods

Abstract

Empirical best linear unbiased prediction (EBLUP) method uses a linear mixed model in combining information from different sources of information. This method is particularly useful in small area problems. The variability of an EBLUP is traditionally measured by the mean squared prediction error (MSPE), and interval estimates are generally constructed using estimates of the MSPE. Such methods have shortcomings like under-coverage or over-coverage, excessive length and lack of interpretability. We propose a parametric bootstrap approach to estimate the entire distribution of a suitably centered and scaled EBLUP. The bootstrap histogram is highly accurate, and differs from the true EBLUP distribution by only $O(d^3n^{-3/2})$, where $d$ is the number of parameters and $n$ the number of observations. This result is used to obtain highly accurate prediction intervals. Simulation results demonstrate the superiority of this method over existing techniques of constructing prediction intervals in linear mixed models.