On robust stopping times for detecting changes in distribution - Yuri Golubev; Mher M. Safarian (Karlsruher Institut für Technologie (KIT), Institut für Volkswirtschaftslehre (ECON), 2018)
Let X1,X2,… be independent random variables observed sequentially and such that X1,…,X-1 have a common probability density p0, while X,X+1,… are all distributed according to p1≠p0. It is assumed that p0 and p1 are known, but the time change ∈Z+ is unknown and the goal is to construct a stopping time τ that detects the change-point as soon as possible. The existing approaches to this problem rely essentially on some a priori information about . For instance, in Bayes approaches, it is assumed that 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 , but have nearly Bayesian detection delays. More precisely, we propose stopping times solving approximately the following problem: Δ(;τα)→minτα subject to α(;τα)≤α for any ≥1, where α(;τ)=Pτ< is 'the false alarm probability' and Δ(;τ)=E(τ-)+ is 'the average detection delay', and explain why such stopping times are robust w.r.t. a priori information about .
urn:nbn:de:swb:90-832796
Keywords: 330; stopping time; false alarm probability; average detection delay; Bayes stopping time; CUSUM method; multiple hypothesis testing