On Optimum Methods in Quickest Detection Problems

Abstract

In this paper optimum methods are developed for observing a process (1), in which the moment when a “disorder” $\theta$ appears is not known. The basic quantity characterizing the quality of this observation method is the mean time delay ${\boldsymbol \tau}$ for detection of a disorder. After making assumption (4) it is shown that for a given false alarm probability $\omega$ or for a given ${\bf N}$ — mathematical expectation of false alarm numbers occurring up to the moment the disorder appears — the observation method minimizing ${\boldsymbol \tau } = {\boldsymbol \tau } (\omega)$ or ${\boldsymbol \tau} = {\boldsymbol \tau} ({\boldsymbol N})$ is based on an observation of a posteriors probability (23). In § 3 a case is considered, wherein, the disorder appears on the background of steadystate conditions arising when the disorder is absent. A method is found for minimizing ${\boldsymbol \tau} = {\boldsymbol \tau} ({\bf T})$ for a set ${\bf T}$ — mathematical expectation of the time between two false alarms. The dependence ${\boldsymbol \tau} = {\boldsymbol \tau} ({\bf T})$ is given by formula (36).