Abstract: For estimation of a -variate mean vector based on a random sample of size drawn from a distribution of a location family, a generalized Stein estimator may be defined which shrinks the sample mean towards a proper linear subspace of . In general, the conventional parametric bootstrap consistently estimates the limit distribution of when , but fails to be consistent otherwise. We establish consistency of two modified forms of the parametric bootstrap for any , which are therefore useful for statistical inference about . In the context of constructing confidence sets for , we show that the first approach, which is based on the out of bootstrap, yields coverage error of order for all , provided that the bootstrap resample size 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 and of order for . 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.
Key words and phrases: Confidence set, consistency, coverage error, iterated bootstrap, m out of n parametric bootstrap, minimax, Stein estimator.