On a class of m out of n bootstrap confidence intervals

It is widely known that bootstrap failure can often be remedied by using a technique known as the 'm out of n' bootstrap, by which a smaller number, m say, of observations are resampled from the original sample of size n, In successful cases of the bootstrap, the m out of n bootstrap is often deemed unnecessary. We show that the problem of constructing nonparametric confid ence intervals is an exceptional case. By considering a new class of m out of n bootstrap confidence limits, we develop a computationally efficient ap proach based on the double bootstrap to construct the optimal m out of n bo otstrap intervals. We show that the optimal intervals have a coverage accur acy which is comparable with that of the classical double-bootstrap interva ls, and we conduct a simulation study to examine their performance. The res ults are in general very encouraging. Alternative approaches which yield ev en higher order accuracy are also discussed.