Sequential Change-Point Problems

Abstract

In this chapter, we consider the problem of sequential change-point detection. Some well-known sequential methods have already been discussed in Chapter 2: the CUSUM test, the GRSh method, Shewhart’s chart and the exponential smoothing method. All these methods are parametric, i.e. they use a priori information on distributions of a random sequence and/or a change-point. However, it turned out that a nonparametric version can be proposed for anyone of them. Our main goal in this chapter is to compare these nonparametric versions. Our approach to a comparative analysis is based upon the fact that, for most of the sequential change-point detection methods, a “large” parameter N can be proposed, a quantity which when tends to infinity, the probability of “false alarm” tends to zero under relatively broad assumptions, and the “delay time” normalized on N tends to a certain determined limit that can be computed for any method of detection. Besides, there exists an apriori low boundary for this limit and different methods of change-point detection can be compared on the basis of their attainment of, or their proximity to, this boundary, i.e. by the degree of realization of their potential properties.