Estimated sampling distributions: the bootstrap and competitors

Let X1,X2,⋯,Xn be i.i.d random variables with d.f.F. Suppose the {^Tn=^Tn(X1,X2,⋯,Xn);n≥1} are real-valued statistics and the {Tn(F);n≥1} are centering functionals such that the asymptotic distribution of n1/2{^Tn−Tn(F)} is normal with mean zero. Let Hn(x,F) be the exact d.f. of n1/2{^Tn−Tn(F)}. The problem is to estimate Hn(x,F) or functionals of Hn(x,F). Under regularity assumptions, it is shown that the bootstrap estimate Hn(x,^Fn), where ^Fn is the sample d.f., is asymptotically minimax; the loss function is any bounded monotone increasing function of a certain norm on the scaled difference n1/2{Hn(x,^Fn)−Hn(x,F)}. The estimated first-order Edgeworth expansion of Hn(x,F) is also asymptotically minimax and is equivalent to Hn(x,^Fn) up to terms of order n−1/2. On the other hand, the straightforward normal approximation with estimated variance is usually not asymptotically minimax, because of bias. The results for estimating functionals of Hn(x,F) are similar, with one notable difference: the analysis for functionals with skew-symmetric influence curve, such as the mean of Hn(x,F), involves second-order Edgeworth expansions and rate of convergence n−1.