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, , X-₁ are iid distributed according to a known distribution F₀ and X_, X+₁, are iid distributed according to a known distribution F₁. Assume that 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 - N) subject to a restriction E (N =) 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^ =) = B for which equation*split₁ < E (N^ - N^) \\ - \ₒₓ₎₏₏₈₍₆ ₑₔ₋₄ₒ ₍|₄ (₍| =) ₁\ ₁ < E (N - N) = o (1) splitequation* where o (1) 0 as B.