Optimal Detection of a Change in Distribution

Abstract

Suppose one is able to observe sequentially a series of independent observations Xl, X2, *-- such that Xi, X2, *-.., X,v1 are iid distributed according to a known distribution Fo and X,, X,,+, *-. are iid distributed according to a known distribution Fl. Assume that v is unknown and the problem is to raise an alarm as soon as possible after the distribution changes from Fo to Fl. Formally, the problem is to find a stopping rule N which in some sense minimizes E(N - I l N 2 v) subject to a restriction E(N I v = mo) 2 B. A stopping rule that is a limit of Bayes rules is first derived. Then an almost minimax rule is presented; i.e. a stopping rule N* is described which satisfies E(N* I v = Xo) = B for which supl,,