Estimated Sampling Distributions: The Bootstrap and Competitors

Let X₁, X₂, ⋯, Xₙ be i.i.d random variables with d.f. F. Suppose the ̂ₙ = T̂ₙ(X₁, X₂, ⋯, Xₙ); n ≥ 1\ are real-valued statistics and the ₙ(F); n ≥ 1\ are centering functionals such that the asymptotic distribution of n1/2̂ₙ - Tₙ(F)\ is normal with mean zero. Let Hₙ(x, F) be the exact d.f. of n1/2̂ₙ - 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, F̂ₙ), where F̂ₙ 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 n1/2ₙ(x, F̂ₙ) - Hₙ(x, F)\. The estimated first-order Edgeworth expansion of Hₙ(x, F) is also asymptotically minimax and is equivalent to Hₙ(x, F̂ₙ) 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.