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\noindent Statistica Sinica {\bf 16}(2006), 45-75

\vspace{3pc} \normalsize
\centerline{\large\bf STEIN CONFIDENCE SETS BASED ON NON-ITERATED}
\vspace{2pt}
\centerline{\large\bf  AND ITERATED PARAMETRIC BOOTSTRAPS}
\vspace{.25cm}
\author{K. Y. Cheung$^1$, Stephen M. S. Lee$^1$ and G. Alastair Young$^2$}
\vspace{.4cm}
\centerline{\it $^1$The University of Hong Kong and $^2$Imperial College London}
%University of Cambridge}
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\begin{quotation}
\noindent {\it Abstract:}
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
$T_{n,S}$ may be defined which shrinks the sample mean towards a proper linear subspace ${\mathbb L}$ of ${\mathbb R}^d$.
In general, the conventional parametric bootstrap consistently estimates the limit distribution
of $n^{1/2}(T_{n,S}-\theta)$ when $\theta\not\in {\mathbb L}$, but fails to be consistent
otherwise.
We establish consistency of two modified forms of the parametric bootstrap for any $\theta\in{\mathbb R}^d$, 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 $\theta\in {\mathbb L}$ and of order $O(n^{-1})$ for $\theta\not\in {\mathbb L}$.
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.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Confidence set, consistency, coverage error, iterated bootstrap,
$m$ out of $n$ parametric bootstrap, minimax, Stein estimator.\par
\end{quotation}

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