Nonparametric Stopping Rules for Detecting Small Changes in Location and Scale Families

Nonparametric analogues of the Page-CUSUM procedure are constructed for sequential detection of location change in a distribution known to be symmetric about 0 and for detecting location change or scale change in an arbitrary unknown distribution. These stopping rules are defined on doubly-indexed stochastic processes whose weak limits are derived when there is no change and when there is a contiguous change. New fluctuation inequalities for rank sums are derived for proving tightness of these processes. In terms of these convergence properties, the nonparametric stopping rules are asymptotically equivalent to their parametric counterparts if the score functions used in both procedures are appropriate for the true density, but even otherwise, the nonparametric rules maintain their false alarm rates (due to the distribution-free property of ranks in the null case) and have good detection properties. The weak convergence results also show how the drift terms, which set in after a change occurs, and drive the underlying processes towards the decision boundary, slow down under model misspecification for both the parametric and the nonparametric procedures.