Abstract
Given independent observations X 1n , …, X nn in R s , let [Fcirc](x) be their weighted empirical distribution with weights w 1n , …, w nn . We obtain cumulant expansions for the weighted estimate T([Fcirc]) 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([Fcirc]) 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.
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