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Cornish-Fisher Expansions for Functionals of the Weighted Partial Sum Empirical Distribution

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  • Christopher S. Withers

    (Callaghan Innovation)

  • Saralees Nadarajah

    (University of Manchester)

Abstract

Given a random sample X1,…,Xn in ℝ p $\mathbb {R}^{p}$ from some distribution F and real weights w1, n,…,wn, n adding to n, define the weighted partial sum empirical distribution as G n ( x , t ) = n − 1 ∑ i = 1 [ n t ] w i , n I X i ≤ x $$ \begin{array}{@{}rcl@{}} \displaystyle G_{n} (\textbf{x}, t) = n^{-1} \sum\limits_{i=1}^{[nt]} w_{i, n} I \left( \textbf{X}_{i} \leq \textbf{x} \right) \end{array} $$ for x in ℝ p $\mathbb {R}^{p}$ , 0 ≤ t ≤ 1. We give Cornish-Fisher expansions for smooth functionals of Gn, following up on Withers and Nadarajah (Statistical Methodology 12:1–15, 2013) who gave expansions for the unweighted version. Applications to sequential analysis include weighted cusum-type functionals for monitoring variance, and a Studentized weighted cusum-type functional for monitoring the mean.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09894-2
    DOI: 10.1007/s11009-021-09894-2
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    References listed on IDEAS

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    1. 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.
    2. N. Mukhopadhyay & T. Solanky, 1997. "Estimation after sequential selection and ranking," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 95-106, January.
    3. 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.
    4. Beirlant, Jan & Vanmarcke, Ann, 1992. "A note on Bahadur-Kiefer-type expansions for the inverse empirical Laplace transform," Statistics & Probability Letters, Elsevier, vol. 15(4), pages 305-311, November.
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