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.