Bootstrap under nonstandard conditions

Abstract

M-estimation under non-standard conditions often yields M-estimators converging weakly at rates different from n^1/2 with typically non-Gaussian weak limits. Although m out of n bootstrap has been shown to estimate the sampling distributions of M-estimators consistently under very general conditions, it suffers from two drawbacks. First, there is as yet no simple routine for choosing the suitable bootstrap sample size m. Secondly, when applied to construct confidence intervals, the m out of n bootstrap typically requires explicit knowledge of the convergence rate n^alpha of the M-estimator, whereas the classical n out of n bootstrap does not require such knowledge. We investigate empirically the weighted bootstrap as a plausible alternative to m out of n bootstrap. Of particular interest is whether the use of non-uniform weights can alleviate the inconveniences posed by the m out of n bootstrap.