Importance Sampling for Bootstrap Confidence Intervals

Abstract

The use of importance-sampling methods to substantially reduce the amount of resampling necessary for the construction of nonparametric bootstrap confidence intervals is investigated. The classical importance-sampling method of Hammersley and Handscomb (1964) is modified to apply to the estimation of quantiles. Based on this method, a resampling procedure is introduced that recenters the bootstrap distribution of a robust estimator of location to produce estimated confidence limits that are much more accurate (for a given amount of resampling) than those obtained by the usual bootstrap method (Efron 1982). The required recentering is accomplished with a suitable “exponential tilting” similar to that used in another connection by Field and Hampel (1982). This importance-sampling procedure is used to produce bootstrap confidence intervals for location based on a class of estimators that includes symmetric M estimators. These interval estimates are asymptotically optimal in a certain sense. Simulation results for 95% confidence intervals based on a particular robust one-step (from the median) M estimator and its studentized version are presented for sample size 20 for the normal and the normal/uniform (slash) distributions. Studentization is accomplished by dividing the location estimator by the sample analog of its asymptotic standard deviation. The importance-sampling results obtained for bootstrap replication sizes 10, 20, and 40 are compared with the standard bootstrap confidence-interval method based on 400 replications. The simulation size for each case is 10,000. These results suggest that the bootstrap replication size may be reduced by as much as a factor of 10 or more when importance sampling is used. The proposed confidence-interval procedures provide alternatives to those given by Gross (1976), Davison and Hinkley (1986), and DiCiccio and Tibshirani (1987). It seems likely that the general approach proposed in this article could be useful in a variety of other confidence-interval estimation problems.

Key Words:

• M estimators
• Quantile estimation
• Resampling methods
• Robust estimation of location
• Variance reduction