Parametric bootstrap inference for stratified models with high-dimensional nuisance specifications

TL;DR

This paper demonstrates that parametric bootstrap methods provide highly accurate inference for stratified models with many nuisance parameters, matching analytical modifications' performance even as the number of strata grows rapidly.

Contribution

It proves that parametric bootstrap inference is as effective as analytical modifications of likelihood pivots in high-dimensional stratified models with many nuisance parameters.

Findings

Bootstrap achieves high accuracy in extreme scenarios.

Equivalence holds regardless of bootstrap constraint type.

Performance remains robust as the number of strata increases.

Abstract

Inference about a scalar parameter of interest typically relies on the asymptotic normality of common likelihood pivots, such as the signed likelihood root, the score and Wald statistics. Nevertheless, the resulting inferential procedures are known to perform poorly when the dimension of the nuisance parameter is large relative to the sample size and when the information about the parameters is limited. In many such cases, the use of asymptotic normality of analytical modifications of the signed likelihood root is known to recover inferential performance. It is proved here that parametric bootstrap of standard likelihood pivots results in as accurate inferences as analytical modifications of the signed likelihood root do in stratified models with stratum specific nuisance parameters. We focus on the challenging case where the number of strata increases as fast or faster than the stratum samples size. It is also shown that this equivalence holds regardless of whether constrained or unconstrained bootstrap is used. This is in contrast to when the number of strata is fixed or increases slower than the stratum sample size, where we show that constrained bootstrap corrects inference to a higher order than unconstrained bootstrap. Simulation experiments support the theoretical findings and demonstrate the excellent performance of bootstrap in extreme scenarios.