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 X_ν, 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(Nν = ∞) ≥ 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 {equation*}{split}1≤ν < ∞E(N^ - ν N^ ≥ ν) \\ - inf_{\{stopping rules N|E(N| ν=∞)≥ B\}} 1≤ν < ∞E(N - ν N ≥ ν) = o(1){split}{equation*} where o(1) → 0 as B → ∞.