Smoothed Bootstrap Percentile Ellipsoidal Confidence Region for Mean Vector

• Activity Number: 462
• Type: Contributed
• Date/Time: Wednesday, August 1, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Santu Ghosh*+ and Alan Polansky
• Companies: Northern Illinois University and Northern Illinois University
• Keywords: Bootstrap percentile method ; Bandwidth ; Confidence Region ; Edgeworth expansion

Abstract

Some studies of the bootstrap have assessed the effect of smoothing the estimated distribution that is resampled, a process usually known as the smoothed bootstrap. Generally, the smoothed distribution of resampling is a kernel estimate, and is often rescaled to retain certain characteristics of the empirical distribution. It has been shown that under the smooth function model, proper bandwidth selection can accomplish a first order-correction for the one-sided univariate percentile interval. In this work we extend these findings to the multivariate case. In univariate case, only an interval is considered as confidence region while in the multivariate case, there are a much richer variety of convex sets. Here we concentrate only on ellipsoidal regions for unknown population mean vector. It is already known that bootstrap percentile ellipsoidal confidence region is second-order accurate. It is shown that with proper selection of bandwidth matrix, one can achieve fourth-order accuracy for the bootstrap percentile ellipsoidal confidence region using the smoothed bootstrap. With the objective of reducing the coverage error, the appropriate bandwidth matrix converges at certain rate.