Statistica Sinica 16(2006), 45-75

STEIN CONFIDENCE SETS BASED ON NON-ITERATED

AND ITERATED PARAMETRIC BOOTSTRAPS

K. Y. Cheung$^1$, Stephen M. S. Lee$^1$ and G. Alastair Young$^2$

$^1$The University of Hong Kong and $^2$Imperial College London

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.

Key words and phrases: Confidence set, consistency, coverage error, iterated bootstrap, m out of n parametric bootstrap, minimax, Stein estimator.