2018/04/24 by Yuri Golubev, Golubev, Yuri, Mher Safarian +1
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1804.09014
arxiv created 2018/04/24 · arxiv updated 2018/04/25
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: \beginsplit Δ(θ;τα)→minτα subject to α(θ;τα)≤ α for any θ≥1, \endsplit where α(θ;τ)=Pθ\τ<θ\ is the false alarm probability and Δ(θ;τ)=Eθ(τ-θ)+ is the average detection delay, %In this paper, we construct \widetildeτα such that %% maxθ≥ 1α(θ;\widetildeτα)≤ α and %Δ(θ;\widetildeτα)≤ (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.