Optimal bootstrap sample size in construction of percentile confidence bounds (Q2722313)

Let \({\mathbf x}=(x_{1},\dots,x_{n})\) be a random sample drawn from an unknown \(d\)-variate distribution \(F.\) An upper \(\alpha\)-level confidence bound for a scalar parameter \(\theta=\theta(F)\) can be obtained by using the bootstrap approach [\textit{B. Efron}, The jackknife, the bootstrap and other resampling plans. (1982; Zbl 0496.62036)]. Its practical implementation consists of drawing with replacement lots of bootstrap samples, each of size \(n,\) from \({\mathbf x}.\) The empirical distribution of the bootstrap samples is then used to approximate the sampling distribution of a statistic designed to link up \({\mathbf x}\) and \(\theta.\) \textit{J.W.H. Swanepoel} [Commun. Stat., Theory Methods 15, 3193-3203 (1986; Zbl 0623.62041)] showed that using a bootstrap sample size different from \(n\) may sometimes be beneficial, especially in cases where Efron's traditional bootstrap is known to fail. In particular, Swanepoel found from a simulation study that a bootstrap sample size of \(2n/3\) was effective in reducing coverage error of symmetric percentile-\(t\) intervals. \textit{P.J. Bickel, F. Goetze} and \textit{W.R. van Zwet} [Stat. Sin. 7, No. 1, 1-31 (1997; Zbl 0927.62043)] reconsidered this approach, among others, in a general framework and referred to it as the \(m\) out of \(n\) with replacement bootstrap, or simply the \(m/n\) bootstrap, where \(m\) denotes the bootstrap sample size. They established an \(n^{1/2}\) consistent estimator based on the \(m/n\) bootstrap and an extrapolation technique.NEWLINENEWLINENEWLINEIn this paper use is made of the \(m/n\) bootstrap to correct for coverage error bootstrap percentile method confidence bounds. Two versions of the bootstrap percentile method, namely the backward percentile and the hybrid percentile, form the basis of the new approach. It is shown that the coverage error of a bootstrap percentile method confidence bound, which is of order \(O(n^{1/2})\) typically, can be reduced to \(O(n^{-1})\) by use of an optimal bootstrap sample size. A simulation study is conducted to illustrate these findings, which also suggest that the new method yields intervals of shorter length and greater stability compared to competitors of similar coverage accuracy.