Asymptotically Optimal Change Point Detection for Composite Hypothesis in State Space Models

This paper investigates change point detection in state space models, in which the pre-change distribution <inline-formula> <tex-math notation="LaTeX">$f^{\theta _{0}}$ </tex-math></inline-formula> is given, while the post-distribution <inline-formula> <tex-math notation="LaTeX">$f^{\theta }$ </tex-math></inline-formula> after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from <inline-formula> <tex-math notation="LaTeX">$f^{\theta _{0}}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$f^{\theta }$ </tex-math></inline-formula>, under a restriction on the false alarms. We investigate theoretical properties of a weighted Shiryayev-Roberts-Pollak (SRP) change point detection rule in state space models. By making use of a Markov chain representation for the likelihood function, exponential embedding of the induced Markovian transition operator, nonlinear Markov renewal theory, and sequential hypothesis testing theory for Markov random walks, we show that the weighted SRP procedure is second-order asymptotically optimal. To this end, we derive an asymptotic approximation for the expected stopping time of such a stopping scheme when the change time <inline-formula> <tex-math notation="LaTeX">$\omega = 1$ </tex-math></inline-formula>. To illustrate our method we apply the results to two types of state space models: general state Markov chains and linear state space models. Simulation study is given to illustrate the method.