Optimal and Asymptotically Optimal CUSUM Rules for Changepoint Detection in the Brownian Motion Model with Multiple Alternatives

• Activity Number: 429
• Type: Contributed
• Date/Time: Thursday, August 12, 2004 : 10:30 AM to 12:20 PM
• Sponsor: Section on Quality and Productivity
• Author(s): Olympia Hadjiliadis*+ and George V. Moustakides
• Companies: Columbia University and INRIA
• Keywords: 2-CUSUM ; changepoint detection ; ARL ; multiple alternatives

Abstract

Statistical surveillance finds application in different areas such as quality control, computer networks, monitoring of health events, etc. This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a change in the constant drift by means of a stopping rule when there are multiple possibilities for such a change. As a performance measure an extended Lorden's criterion is proposed. The goal is to minimize the worst case detection delay subject to a false alarm constraint on the in-control ARL. When the drifts have the same sign, the CUSUM rule designed to detect the smallest in absolute value drift, is proven to be optimum. If the drifts have opposite signs then a specific 2-CUSUM rule is shown to be asymptotically optimal as the in-control ARL tends to infinity. In particular, when the drifts are equal in absolute value, the difference in performance between the unknown optimal rule and the proposed scheme remains uniformly bounded, although both quantities tend to infinity; for unequal in absolute value drifts the asymptotic optimality is even stronger since the corresponding difference tends to zero.