Asymptotic operating characteristics of an optimal change point detection in hidden Markov models

Let \xi_0,\xi_1,...,\xi_{\omega-1} be observations from the hidden Markov model with probability distribution P^{\theta_0}, and let \xi_{\omega},\xi_{\omega+1},... be observations from the hidden Markov model with probability distribution P^{\theta_1}. The parameters \theta_0 and \theta_1 are given, while the change point \omega is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from P^{\theta_0} to P^{\theta_1}, but to avoid false alarms. Specifically, we seek a stopping rule N which allows us to observe the \xi's sequentially, such that E_{\infty}N is large, and subject to this constraint, sup_kE_k(N-k|N\geq k) is as small as possible. Here E_k denotes expectation under the change point k, and E_{\infty} denotes expectation under the hypothesis of no change whatever. In this paper we investigate the performance of the Shiryayev-Roberts-Pollak (SRP) rule for change point detection in the dynamic system of hidden Markov models. By making use of Markov chain representation for the likelihood function, the structure of asymptotically minimax policy and of the Bayes rule, and sequential hypothesis testing theory for Markov random walks, we show that the SRP procedure is asymptotically minimax in the sense of Pollak [Ann. Statist. 13 (1985) 206-227]. Next, we present a second-order asymptotic approximation for the expected stopping time of such a stopping scheme when \omega=1. Motivated by the sequential analysis in hidden Markov models, a nonlinear renewal theory for Markov random walks is also given.