A Robust Surveillance Scheme for Stochastically Ordered Alternatives

Abstract

We sequentially observe independent observations $X_1, X_2,\ldots$ such that initially they have distribution $G_0$; at some unknown time $\nu$ they become stochastically larger, having distribution $G_1$. Neither $G_0$ nor $G_1$ is fully specified. We wish to detect that a change has taken place as soon as possible after its occurrence, subject to a constraint on the rate of false alarms. We derive a family of nonparametric sequential procedures based on ranks, with noncontiguous alternatives in mind. Large-sample approximations to the operating characteristics are obtained analytically. The proposed procedures all possess robustness of validity, because they are based on ranks. Near-optimal sensitivity can be obtained for specific alternatives by choosing an appropriate procedure. For example, when observations are normally distributed with one standard deviation shift in mean postchange, an appropriate nonparametric surveillance scheme yields 97% asymptotic relative efficiency, as compared to the optimal procedure when all distributional parameters are known. Our procedures are computationally feasible. Monte Carlo experiments confirm the applicability of the asymptotic calculations, including high levels of efficiency, for sample sizes met in practice.