Parametric Bootstrap Inference for Stratified Models With High-Dimensional Nuisance Specifications

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 have been 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 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 dimension of the parameter space is fixed relative to the sample size, where constrained bootstrap is known to correct inference to higher-order than unconstrained bootstrap does. Large scale simulation experiments support the theoretical findings and demonstrate the excellent performance of bootstrap in some extreme modelling scenarios.