Stein confidence sets based on non-iterated and iterated parametric bootstraps

Abstract

For estimation of a d-variate mean vector 6 based on a random sample of size n drawn from a distribution of a location family, a generalized Stein estimator T n,S may be defined which shrinks the sample mean towards a proper linear subspace double-struck L sign of ℝ d. In general, the conventional parametric bootstrap consistently estimates the limit distribution of n 1/2 (T n,S - θ) when θ ∉ double-struck L sign but fails to be consistent otherwise. We establish consistency of two modified forms of the parametric bootstrap for any θ ∈ ℝ d, which are therefore useful for statistical inference about 9. In the context of constructing confidence sets for θ, we show that the first approach, which is based on the m out of n bootstrap, yields coverage error of order O(n -1/4) for all θ, provided that the bootstrap resample size m has an order determined by a minimax criterion. The second approach bootstraps from a distribution with an adaptively estimated mean vector, and is shown to yield coverage error of exponentially small order for θ ∈ double-struck L sign and of order O(n -1) for θ ∉ double-struck L sign. Iterated versions of the two approaches are also developed to give improved orders of coverage error. A simulation study is reported to illustrate our asymptotic findings.