Kullback-Leibler Approach to CUSUM Quickest Detection Rule for Markovian Time Series

Valerie Girardin (1) , Victor Konev (2) , Serguei Pergamenchtchikov (3, 2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 Tomsk State University [Tomsk]
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

Optimality properties of decision procedures are studied for the quickest detection of a change-point of parameters in autoregressive and other Markov type sequences. The limit of the normalized conditional log-likelihood ratios is shown to exist for Markov chains satisfying the ergodic theorem of information theory. Then closed-form expressions for this limit are derived by making use of the time average rate of Kullback-Leibler divergence. The good properties of the detection procedures based on a sequential analysis approach are proven to hold thanks to geometric ergodicity properties of the observation processes. In particular, the window-limited CUSUM rule is shown to be optimal for detecting the disruption point in autore-gressive models. Sparre Andersen models are specifically studied.

Mots clés: Change-point detection; CUSUM detection procedure; Ergodic Theorem of information theory; Kullback-Leibler divergence; Relative entropy; Sequential detection; Sparre Andersen models; Kullback- Leibler divergence

Domaines: Statistiques [math.ST]

Valerie Girardin, Victor Konev, Serguei Pergamenchtchikov. Kullback-Leibler Approach to CUSUM Quickest Detection Rule for Markovian Time Series. Sequential Analysis, 2018, 37, pp.322 - 341. ⟨10.1080/07474946.2018.1548846⟩. ⟨hal-02334914⟩