High-order Coverage of Smoothed Bayesian Bootstrap Intervals for Population Quantiles

Abstract

We characterize the high-order coverage accuracy of smoothed and unsmoothed Bayesian bootstrap confidence intervals for population quantiles. Although the original (Rubin 1981) unsmoothed intervals have the same O(n−1/2) coverage error as the standard empirical bootstrap, the smoothed Bayesian bootstrap of Banks (1988) has much smaller O(n−3/2[log(n)]3) coverage error and is exact in special cases, without requiring any smoothing parameter. It automatically removes an error term of order 1/n that other approaches need to explicitly correct for. This motivates further study of the smoothed Bayesian bootstrap in more complex settings and models.