Optimal detection of a change in distribution

Suppose one is able to observe sequentially a series of independent observations X1,X2,⋯ such that X1,X2,⋯,Xν−1 are iid distributed according to a known distribution F0 and Xν,Xν+1,⋯ are iid distributed according to a known distribution F1. Assume that ν is unknown and the problem is to raise an alarm as soon as possible after the distribution changes from F0 to F1. Formally, the problem is to find a stopping rule N which in some sense minimizes E(N−ν∣N≥ν) subject to a restriction E(N∣ν=∞)≥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∗∣ν=∞)=B for which sup1≤ν<∞E(N∗−ν∣N∗≥ν)−inf{stopping rulesN|E(N|ν=∞)≥B}sup1≤ν<∞E(N−ν∣N≥ν)=o(1) where o(1)→0 as B→∞.