Statistica Sinica 18(2008), 967-985

ON THE CHOICE OF $\mbox{\boldmath $m$}$ IN THE $\mbox{\boldmath $m$}$

OUT OF $\mbox{\boldmath $n$}$ BOOTSTRAP AND

CONFIDENCE BOUNDS FOR EXTREMA

Peter J. Bickel and Anat Sakov

University of California, Berkeley and Tanor Group

Abstract: For i.i.d. samples of size $n$, the ordinary bootstrap (Efron (1979)) is known to be consistent in many situations, but it may fail in important examples (Bickel, Götze and van Zwet (1997)). Using bootstrap samples of size $m$, where $m\rightarrow\infty$ and $m/n\rightarrow 0$, typically resolves the problem (Bickel et al. (1997), Politis and Romano (1994)). The choice of $m$ is a key issue. In this paper, we consider an adaptive rule, proposed by Bickel, Götze, and van Zwet (personal communication), to pick $m$. We give general sufficient conditions for first order validity of the rule, and consider its higher order behavior when the ordinary bootstrap fails, and when it works. We then examine the behavior of the rule in the context of setting confidence bounds on high percentiles, such as the asymptotic expected maximum.

Key words and phrases: Adaptive choice, bootstrap, choice of m, data-dependent rule, extrema, m out of n bootstrap.