Asymptotic Optimality of Mixture Rules for Detecting Changes in General Stochastic Models

TL;DR

This paper extends the asymptotic optimality theory of mixture-based sequential change detection rules to general stochastic models with unknown post-change parameters, demonstrating their effectiveness in complex, dependent data scenarios.

Contribution

It introduces mixture Shiryaev and Shiryaev-Roberts rules for non-i.i.d. models with unknown post-change parameters and proves their asymptotic optimality.

Findings

Mixture rules are asymptotically optimal under certain conditions.

The paper generalizes previous i.i.d. results to dependent, non-i.i.d. models.

Provides theoretical foundations for practical change detection in complex stochastic systems.

Abstract

The paper addresses a sequential changepoint detection problem for a general stochastic model, assuming that the observed data may be non-i.i.d. (i.e., dependent and non-identically distributed) and the prior distribution of the change point is arbitrary. Tartakovsky and Veeravalli (2005), Baron and Tartakovsky (2006), and, more recently, Tartakovsky (2017) developed a general asymptotic theory of changepoint detection for non-i.i.d.\ stochastic models, assuming the certain stability of the log-likelihood ratio process, in the case of simple hypotheses when both pre-change and post-change models are completely specified. However, in most applications, the post-change distribution is not completely known. In the present paper, we generalize previous results to the case of parametric uncertainty, assuming the parameter of the post-change distribution is unknown. We introduce two detection rules based on mixtures -- the Mixture Shiryaev rule and the Mixture Shiryaev--Roberts rule -- and study their asymptotic properties in the Bayesian context. In particular, we provide sufficient conditions under which these rules are first-order asymptotically optimal, minimizing moments of the delay to detection as the probability of false alarm approaches zero.