On robust stopping times for detecting changes in distribution

Let X1,X2,… be independent random variables observed sequentially and such that X1,…,Xθ−1 have a common probability density p0, while Xθ,Xθ+1,… are all distributed according to p1≠p0. It is assumed that p0 and p1 are known, but the time change θ∈Z+ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The existing approaches to this problem rely essentially on some a priori information about θ. For instance, in Bayes approaches, it is assumed that θ is a random variable with a known probability distribution. In methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times which do not make use of a priori information about θ, but have nearly Bayesian detection delays. More precisely, we propose stopping times solving approximately the following problem: Δ(θ;τα)→minτα subject to α(θ;τα)≤α for any θ≥1, where α(θ;τ)=Pθ{τ<θ} is \textit{the false alarm probability} and Δ(θ;τ)=Eθ(τ−θ)+ is \textit{the average detection delay}, %In this paper, we construct τ˜α such that % and (1+o(1))log(θ/α), as θ/α and explain why such stopping times are robust w.r.t. a priori information about θ.