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

## TL;DR

This paper develops a theoretically optimal change point detection method for state space models with unknown post-change distributions, demonstrating second-order asymptotic optimality using advanced Markov process techniques.

## Contribution

It introduces a weighted SRP detection rule for state space models and proves its second-order asymptotic optimality under general conditions.

## Key findings

- The weighted SRP procedure is second-order asymptotically optimal.
- Derived an asymptotic approximation for the expected stopping time.
- Applied the method to general and linear state space models.

## Abstract

This paper investigates change point detection in state space models, in which the pre-change distribution $f^{\theta_0}$ is given, while the poster distribution $f^{\theta}$ after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from $f^{\theta_0}$ to $f^{\theta}$, 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 $\omega = 1$. To illustrate our method we apply the results to two types of state space models: general state Markov chains and linear state space models.