Cornish–Fisher expansions for functionals of the partial sum empirical distribution

Given a random sample X1,…,Xn in RpRp from some distribution function FF we define the partial sum empirical distribution function as Gn(x,t)=n−1∑i=1[nt]I(Xi≤x) for x in RpRp, 0≤t≤10≤t≤1. We give Cornish–Fisher expansions for smooth functionals of GnGn. Applications to sequential analysis include cusum-type functionals for monitoring variance, and a Studentized cusum-type functional for monitoring the mean.