2010/02/23 by Christopher S. Withers, C. S. Withers, Withers, C. S. +3
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.ME #stat.TH
paper · pdf · doi:10.48550/arxiv.1002.4338
arxiv created 2010/02/23 · openalex publication_date 2010/02/23 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper extends Edgeworth-Cornish-Fisher expansions for the distribution and quantiles of nonparametric estimates in two ways. Firstly it allows observations to have different distributions. Secondly it allows the observations to be weighted in a predetermined way. The use of weighted estimates has a long history including applications to regression, rank statistics and Bayes theory. However, asymptotic results have generally been only first order (the CLT and weak convergence). We give third order asymptotics for the distribution and percentiles of any smooth functional of a weighted empirical distribution, thus allowing a considerable increase in accuracy over earlier CLT results. Consider independent non-identically distributed (\it non-iid) observations X1n, ..., Xnn in Rs. Let F(x) be their \it weighted empirical distribution with weights w1n, ..., wnn. We obtain cumulant expansions and hence Edgeworth-Cornish-Fisher expansions for T(F) for any smooth functional T(⋅) by extending the concepts of von Mises derivatives to signed measures of total measure 1. As an example we give the cumulant coefficients needed for Edgeworth-Cornish-Fisher expansions to O(n-3/2) for the sample variance when observations are non-iid.