On a new method for constructing bootstrap confidence bounds

Abstract

It is well-known that the standard methods for constructing bootstrap confidence bounds or intervals are in many situations not sufficiently accurate, that is, coverage probabilities converge to the nominal level at unsatisfactory rates. We propose a new method, based on sample splitting, for constructing higher-order accurate bootstrap confidence bounds for a parameter appearing in the regular smooth function model introduced by Bhattacharya and Ghosh (1978). It has been demonstrated by Hall (1986, 1988, 1992) that the well-known percentile-t bootstrap confidence bound 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 percentile bounds have reduced error of O(n-1). 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. 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 also derive similar results for the case where the slope parameter in a linear regression model is of interest, showing 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 (1946) and Hurt (1976), as well as an expression for an Edgeworth polynomial arising in the linear regression setup. The study is concluded with a modest simulation study, which illustrates the behaviour of the new confidence bounds for small to moderate sample sizes