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

Abstract

Given a random sample X₁,…,X_n in $\mathbb {R}^{p}$ from some distribution F and real weights w_{1, n},…,w_{n, 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 G_n, 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.