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

For estimation of a d-variate mean vector theta based on a random sample of size n drawn from a distribution of a location family, a generalized Stein estimator T-n,T-S may be defined which shrinks the sample mean towards a proper linear subspace L of R-d. In general, the conventional parametric bootstrap consistently estimates the limit distribution of n(1/2) (T-n,T-S - 0) when theta is not an element of L, but fails to be consistent otherwise. We establish consistency of two modified forms of the parametric bootstrap for any theta is not an element of R-d, which are therefore useful for statistical inference about theta. In the context of constructing confidence sets for theta, 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 theta, 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 theta is an element of L and of order O(n(-1)) for theta is not an element of L. 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.