On robust stopping times for detecting changes in distribution
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Let X_1,X_2,... be independent random variables observed sequentially and such that X_1,...,X_θ-1 have a common probability density p_0, while X_θ,X_θ+1,... are all distributed according to p_1≠ p_0. It is assumed that p_0 and p_1 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;τ^α)→_τ^α subject to α(θ;τ^α)<α for any θ>1, where α(θ;τ)=P_θ{τ<θ} is the false alarm probability and Δ(θ;τ)=E_θ(τ-θ)_+ is the average detection delay, that and (1+o(1))(θ/α), as θ/α and explain why such stopping times are robust w.r.t. a priori information about θ.
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