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

Abstract

Given a random sample X1,…,Xn in \(\mathbb {R}^{p}\) from some distribution F and real weights w1, n,…,wn, n adding to n, define the weighted partial sum empirical distribution as

$$ \begin{array}{@{}rcl@{}} \displaystyle G_{n} (\textbf{x}, t) = n^{-1} \sum\limits_{i=1}^{[nt]} w_{i, n} I \left( \textbf{X}_{i} \leq \textbf{x} \right) \end{array} $$

for x in \(\mathbb {R}^{p}\), 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.

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Acknowledgements

The authors thank the Editor and the referee for careful reading and comments which greatly improved the paper.

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Correspondence to Saralees Nadarajah.

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Withers, C.S., Nadarajah, S. Cornish-Fisher Expansions for Functionals of the Weighted Partial Sum Empirical Distribution. Methodol Comput Appl Probab (2021). https://doi.org/10.1007/s11009-021-09894-2

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Keywords

  • Cornish-Fisher expansions
  • Functional derivatives
  • Kiefer process
  • Sequential tests

Mathematics Subject Classification (2010)

  • Primary 62F99
  • Secondary 62G20