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

Given a random sample X 1 ,…, X n in 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 G n ( x , t ) = n − 1 ∑ i = 1 [ n t ] w i , n I X i ≤ x for x in 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.