Iterating the m out of n bootstrap for smooth function models with null derivatives

Abstract

The bootstrap provides an attractive approach to nonparametric inference on a scalar parameter θ of interest. In certain situations validity of the conventional n out of n bootstrap may depend on the true value of the parameter. The m out of n bootstrap is well known to produce consistent estimators of sampling distributions in nonregular problems, where the conventional bootstrap breaks down, but is often accompanied by a loss in efficiency as compared to regular applications of the conventional bootstrap. Of theoretical and practical importance is the question of whether iteration can improve the m out of n bootstrap by reducing its asymptotic error. The problem of constructing a confidence interval for a function θ of a population mean under regularity conditions, but with the function having a null derivative at the true population mean, provides an important testing ground for analysis of iteration of the m out of n bootstrap. In this setting substitution estimators are n-consistent with limiting chi-squared type distributions, and the n out of n bootstrap is inconsistent for their sampling distributions. The m out of n bootstrap percentile method interval is, by contrast, found to be consistent, incurring a one-sided coverage error of order O(n −1/2 ) if m is chosen optimally. We propose a new scheme for iterating the m out of n bootstrap to reduce the coverage error further to order O(n −2/3 ), provided that the first-level and second-level bootstrap resample sizes are appropriately chosen. Our new scheme is computationally directly comparable to the conventional bootstrap iterative scheme as would have been used in regular settings. Several numerical examples, including a natural application of bootstrap confidence intervals in a hypothesis testing problem, are presented to motivate our development and illustrate the theoretical findings.