Statistica Sinica 10(2000), 715-729

SEQUENTIAL DETECTION OF SIGNALS WITH KNOWN

SHAPE AND UNKNOWN MAGNITUDE

M. Beibel

Albert-Ludwigs-Universität Freiburg

Abstract: We study a problem of sequential detection in a continuous time change-point model with a transition period. Let $W$ denote a Brownian motion process which has zero drift during the time interval $[0,\nu)$ and drift $\theta h(t - \nu)$ during the time interval $[\nu,\infty)$. Here $h$ is a known deterministic function and $\theta$ and $\nu$ are unknown parameters. The goal is to find a stopping time $T$ of $W$ that stops as soon and as reliably as possible after the change-point $\nu$. We consider stopping rules based on mixtures of likelihoods and show that they are approximately Bayes optimal.

Key words and phrases: Bayes problem, change-point, sequential detection, transition period.