Bootstrap Asymptotics

The bootstrap, introduced by Efron (1979), merges simulation with formal model-based statistical inference. A statistical model for a sample Xn of size n is a family of distributions {Pθ,n : θ ∈ Θ}. The parameter space Θ is typically metric, possibly infinite-dimensional. The value of θ that identifies the true distribution from which Xn is drawn is unknown. Suppose that θ̂n = θ̂n(Xn) is a consistent estimator of θ. The bootstrap idea is: (a) Create an artificial bootstrap world in which the true parameter value is θ̂n and the sample X ∗ n is generated from the fitted model Pθ̂n,n. That is, the conditional distribution of X∗ n, given the data Xn, is Pθ̂n,n. (b) Act as if a sampling distribution computed in the fully known bootstrap world is a trustworthy approximation to the corresponding, but unknown, sampling distribution in the model world. For example, consider constructing a confidence set for a parametric function τ(θ), whose range is the set T . As in the classical pivotal method, let Rn(Xn, τ(θ)) be a specified root, a real-valued function of the sample and τ(θ). Let Hn(θ) be the sampling distribution of the root under the model. The bootstrap distribution of the root is Hn(θ̂n), a random probability measure that can also be viewed as the conditional distribution of Rn(X ∗ n, τ(θ̂n)) given the sample Xn. An associated bootstrap confidence set for τ(θ), of nominal coverage probability β, is then Cn,B = {t ∈ T : Rn(Xn, t) ≤ H −1 n (β, θ̂n)}. The quantile on the right can be approximated, for instance, by Monte Carlo techniques. The intuitive expectation is that the coverage probability of Cn,B will be close to β whenever θ̂n is close to θ. When does the bootstrap approach work? Bootstrap samples are perturbations of the data from which they are generated. If the goal is to probe how a statistical procedure performs on data sets similar to the one at hand, then repeating the statistical procedure on bootstrap samples stands to be instructive. An exploratory rationale for the bootstrap appeals intellectually when empirically supported probability models for the data are lacking. Indeed, the literature on “statistical inference” continues to struggle with an uncritical tendency to view data as a random sample from a statistical model known to the statistician apart