On the asymptotic theory of new bootstrap confidence bounds

Abstract

We propose a new method, based on sample splitting, for constructing bootstrap confidence bounds for a parameter appearing in the regular smooth function model. It has been demonstrated in the literature, for example, by Hall [Ann. Statist. 16 (1988) 927–985; The Bootstrap and Edgeworth Expansion (1992) Springer], that the well-known percentile-$t$ method for constructing bootstrap confidence bounds typically incurs a coverage error of order $O(n^{-1})$, with $n$ being the sample size. Our version of the percentile-$t$ bound reduces this coverage error to order $O(n^{-3/2})$ and in some cases to $O(n^{-2})$. Furthermore, whereas the standard percentile bounds typically incur coverage error of $O(n^{-1/2})$, the new bounds have reduced error of $O(n^{-1})$. In the case where the parameter of interest is the population mean, we derive for each confidence bound the exact coefficient of the leading term in an asymptotic expansion of the coverage error, although similar results may be obtained for other parameters such as the variance, the correlation coefficient, and the ratio of two means. We show that equal-tailed confidence intervals with coverage error at most $O(n^{-2})$ may be obtained from the newly proposed bounds, as opposed to the typical error $O(n^{-1})$ of the standard intervals. It is also shown that the good properties of the new percentile-$t$ method carry over to regression problems. Results of independent interest are derived, such as a generalisation of a delta method by Cramér [Mathematical Methods of Statistics (1946) Princeton Univ. Press] and Hurt [Apl. Mat. 21 (1976) 444–456], and an expression for a polynomial appearing in an Edgeworth expansion of the distribution of a Studentised statistic for the slope parameter in a regression model. A small simulation study illustrates the behavior of the confidence bounds for small to moderate sample sizes.