The Automatic Construction of Bootstrap Confidence Intervals

Abstract

The standard intervals, for example, $\widehat{\theta }\pm 1.96\widehat{\sigma }$ for nominal 95% two-sided coverage, are familiar and easy to use, but can be of dubious accuracy in regular practice. Bootstrap confidence intervals offer an order of magnitude improvement—from first order to second order accuracy. This article introduces a new set of algorithms that automate the construction of bootstrap intervals, substituting computer power for the need to individually program particular applications. The algorithms are described in terms of the underlying theory that motivates them, along with examples of their application. They are implemented in the R package bcaboot. Supplementary materials for this article are available online.

KEYWORDS:

• bca method
• Exponential families
• Nonparametric intervals
• Second-order accuracy

Supplementary Materials

Title:bcaboot—Bias corrected bootstrap confidence intervals. Available on the Comprehensive R Archive Network (CRAN).

Script to reproduce figures:R scripts to reproduce figures in this article (scripts.zip).

Notes

² See the Appendix for how bcajack proceeds if m does not exactly divide n.

Additional information

Funding

Research supported in part by National Science Foundation award DMS 1608182 (Bradley Efron). Research supported in part by the Clinical and Translational Science Award 1UL1 RR025744 for the Stanford Center for Clinical and Translational Education and Research (Spectrum) from the National Center for Research Resources, National Institutes of Health and award LM07033 (Balasubramanian Narasimhan).