Bootstrapping binomial confidence intervals

Abstract

Traditional confidence intervals for the binomial parameter p which are based on the central limit theorem have coverage error of order O(n − 1 2 ) . We obtain two types of bootstrap intervals with improved coverage accuracy. One is based on the Studentized mean and the other on the normalized mean. The methods used are smoothing and bootstrap calibration (Loh, J. Amer. Statist. Assoc. 82 (1987) 155–162; Ann Statist. 16 (1988) 972–976; Statist. Sinica 1 (1991) 477–491). Coverage errors are reduced to O(n − 3 2 s n logn) for one-sided intervals and O( n −2 s n log n) for two-sided intervals, where s n is any sequence of real numbers tending slowly to infinity. A by-product is that the bootstrap approximation to the smoothed Studentized binomial mean attains the same order as that for continuous distributions; this improves a previous result of Lahiri (unpublished manuscript, 1991). We also compare our intervals with intervals based on the saddlepoint approximation formula, which have O(n − 3 2 ) coverage errors. A generalization of the proposed methods to confidence interval construction for the mean of a discrete random variable taking a finite number of values is also considered. Simulation results indicate that the finite-sample properties of the bootstrap intervals are satisfactory.