Edgeworth Expansions For Functions Of Weighted Empirical distributions With Applications To Nonparametric confidence Intervals

摘要

Given independent observations X_(1n).....X_(nn) in R~s, let F(x) be their weighted empirical distribution with weights w_(1n),.....w_(nn). We obtain cumulant expansions for the weighted estimate T(F) for any smooth functional T(·) by extending the concepts of von Mises derivatives to signed measures of total measure 1. From these are derived third-order Edgeworth-Cornish-Fisher expansions for T(F) and confidence intervals for T(F) of third-order accuracy based on the weighted empirical distribution. These results are also extended to samples from k distributions and confidence intervals for functionals of k distributions.