INRIA report.dvi

This work examines the problem of sequential change detection in the constant drift of a Brownian motion in the case of multiple alternatives. As a performance measure an extended Lorden’s criterion is proposed. When the possible drifts, assumed after the change, have the same sign, the CUSUM rule designed to detect the smallest in absolute value drift, is proven to be the optimum. If the drifts have opposite signs then a specific 2CUSUM rule is shown to be asymptotically optimal as the frequency of false alarms tends to infinity. Key-words: Change Detection, Quickest detection, CUSUM, Two-sided CUSUM. This work was published in the proceedings of the Joint Statistical Meeting JSM’2004, Toronto, Canada, Aug. 2004 and has been submitted for publication to the SIAM journal: Theory of Probability and its Applications. † O. Hadjiliadis is a Ph.D student with the Department of Statistics, Columbia University, New York, USA. ‡ G.V. Moustakides, until July 2004, was a senior researcher with INRIA-Rennes. Since August 2004 he is a full time professor with the Department of Computer and Communication Engineering, University of Thessaly, Volos, Greece. E-mail: [email protected]; Website: www.inf.uth.gr/~moustaki/. Procédures CUSUM optimales et asymptotiquement optimal pour la détection des changements dans un model Brownien à des alternatives multiples Résumé : On considère le problem de détection sequentielle d’une motion Brownienne au drift constant dans le cas des alternatives multiples. Comme critère de performance est proposé une extension du critère de Lorden. Quand les drifts possibles sont du même signe, on démontre que la procédure CUSUM qui détecte le drift de valeur absolue minimale est optimale. Si les drifts ont des signes opposés, on démontre qu’une procédure 2-CUSUM specifique est asymptotiquement optimale, quand la fréquence de faux alarmes tend vers l’infini. Mots-clés : Détection des changements, Détection rapide, CUSUM, CUSUM bidirectionnelle. CUSUM rules for multiple alternatives i