Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

We consider the problem of quick detection of abrupt parameter changes in an autoregressive model with Gaussian white noise. The time of change is deterministic but otherwise unknown. In contrast to detection algorithms for i.i.d. models, the analysis of most procedures known in the literature for detecting disruption in autoregressive schemes reduces to only one characteristic, the false alarm rate, while the delay time characteristic is studied only by means of numerical simulation for some specific models. We develop the method proposed by Lai for detecting abrupt changes in stochastic systems with dependent observations. We prove that window-limited CUSUM rule minimizes the asymptotic detection delay risk under appropriate constraint on the probability of false alarm for the autoregressive model provided that pre-change and post-change parameters are known and the process is stable before and after the disruption. Moreover, it is point-wise optimal. To treat the detection problem for autoregression with unknown post-change parameters we prove some extensions of Lai's theorem for the generalized CUSUM procedure. This result allows us to establish the asymptotic minimaxity and point-wise optimality of CUSUM rule by making use of the risk involving additional supremum in parameter over some compact in the stability region of the autoregressive process.