Accurate tests and intervals based on nonlinear cusum statistics

Suppose that we are monitoring incoming observations for a change in mean via a cusum test statistic. The usual nonparametric methods give first and second order approximations for the one- and two-sided cases. Withers and Nadarajah [2009. Accurate tests and intervals based on linear cusum statistics. Statistics and Probability Letters 79, 689-697] showed how to improve the order of these approximations for linear statistics. Here, an extension is provided for nonlinear statistics with non-zero asymptotic variances. This involves development of two calculi analogous to that of von Mises.