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This paper extends Edgeworth-Cornish-Fisher expansions for the distributionand quantiles of nonparametric estimates in two ways. Firstly it allowsobservations to have different distributions. Secondly it allows theobservations to be weighted in a predetermined way. The use of weightedestimates has a long history including applications to regression, rankstatistics and Bayes theory. However, asymptotic results have generally beenonly first order (the CLT and weak convergence). We give third orderasymptotics for the distribution and percentiles of any smooth functional of aweighted empirical distribution, thus allowing a considerable increase inaccuracy over earlier CLT results.  Consider independent non-identically distributed ({\it non-iid}) observations$X_{1n}, ..., X_{nn}$ in $R^s$. Let $\hat{F}(x)$ be their {\it weightedempirical distribution} with weights $w_{1n}, ..., w_{nn}$. We obtain cumulantexpansions and hence Edgeworth-Cornish-Fisher expansions for $T(\hat{F})$ forany smooth functional $T(\cdot)$ by extending the concepts of von Misesderivatives to signed measures of total measure 1. As an example we give thecumulant coefficients needed for Edgeworth-Cornish-Fisher expansions to$O(n^{-3/2})$ for the sample variance when observations are non-iid.

The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions