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Edgeworth expansions for functions of weighted empirical distributions with applications to nonparametric confidence intervals

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  • Christopher Withers
  • Saralees Nadarajah

Abstract

Given independent observations X1n, …, Xnn in Rs, let [Fcirc](x) be their weighted empirical distribution with weights w1n, …, wnn. 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.

Suggested Citation

  • Christopher Withers & Saralees Nadarajah, 2008. "Edgeworth expansions for functions of weighted empirical distributions with applications to nonparametric confidence intervals," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 751-768.
    • Handle: RePEc:taf:gnstxx:v:20:y:2008:i:8:p:751-768
    DOI: 10.1080/10485250802392971
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    1. Naâmane Laïb, 2003. "Non‐Parametric Testing of Conditional Variance Functions in Time Series," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 45(4), pages 461-475, December.
    2. C. Withers, 1988. "Nonparametric confidence intervals for functions of several distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(4), pages 727-746, December.
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    Cited by:

    1. Christopher S. Withers & Saralees Nadarajah, 2022. "Cornish-Fisher Expansions for Functionals of the Weighted Partial Sum Empirical Distribution," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1791-1804, September.
    2. C. S. Withers, 2024. "5th-Order Multivariate Edgeworth Expansions for Parametric Estimates," Mathematics, MDPI, vol. 12(6), pages 1-28, March.
    3. Withers, Christopher S. & Nadarajah, Saralees, 2010. "The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1093-1102, July.

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