Slightly Conservative Bootstrap for Maxima of Sums

Understand

We study the bootstrap for the maxima of the sums of independent random variables, a problem of high relevance to many applications in modern statistics.

• Since the consistency of bootstrap was justified by Gaussian approximation in Chernozhukov et al.
• (2013), quite a few attempts have been made to sharpen the error bound for bootstrap and reduce the sample size requirement for bootstrap consistency.
• In this paper, we show that the sample size requirement can be dramatically improved when we make the inference slightly conservative, that is, to inflate the bootstrap quantile $t_{\alpha}^*$ by a small fraction, e.g.