Bootstrap inference for a class of non-regular estimators

Abstract

We introduce a class of non-regular estimators in arbitrary normed spaces which include estimators for a non-negative mean, for the squared mean, as well as for the Hodges and Stein estimators. In these cases, the nonparametric bootstrap is consistent on all but a small subset of the underlying parameter space. Bootstrap remedies, such as the $m\text{-out-of-}n$ bootstrap and the oracle bootstrap, have been proposed to mainly solve the inconsistency of the nonparametric bootstrap under a fixed parameter setting. We study the local asymptotic behavior of the estimators and of their bootstrap distributions by allowing the underlying parameter to approach a fixed value at various rates. Our theoretical results determine the precise limiting behavior of the estimators and of their bootstrap distributions in these problems. Simulation results examining the finite sample local performance of the bootstrap estimators are provided.