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

Christopher S. Withers () and Saralees Nadarajah ()
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Christopher S. Withers: Callaghan Innovation
Saralees Nadarajah: University of Manchester

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 3, 1791-1804

Abstract: 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.

Keywords: Cornish-Fisher expansions; Functional derivatives; Kiefer process; Sequential tests; Primary 62F99; Secondary 62G20 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-021-09894-2

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