A note on proving that the (modified) bootstrap works

Abstract

Let be a sample of independent, identically distributed (i.i.d.) random variables with common distribution function F and suppose is a bootstrap sample of i.i.d. random variables from the empirical distribution function (e.d.f.) Fn of . The twofold aim of this paper consists in, firstly providing examples illustrating the fact that the custo-mary choice of m = n is frequently wrong. In fact, specifying m as some suitable function of n it can. among other things, be shown chat the bootstrap also works for the well-known counter-examples given by Bickel and Freedman (1981). Secondly, a method is suggested which can be used to show that the bootstrap method of distribution approximation is asymptotically valid. This method is based on the oscillation behavior of empirical processes rather than the equicontinuity arguments of Bickel and Freedman (1981) which are based on the Mallows metric.