A Numerical Approach to Performance Analysis of Quickest Change-Point Detection Procedures

TL;DR

This paper introduces a numerical method for analyzing the performance of popular quickest change-point detection procedures, enabling precise computation of metrics like detection delay and false alarm time, and explores optimal initialization strategies.

Contribution

It develops integral equations and a numerical approach to evaluate key performance metrics for CUSUM, EWMA, and Shiryaev-Roberts tests, including their initialization strategies and optimality properties.

Findings

Numerical computation of performance metrics is feasible for key detection procedures.

Deterministic initializations can match or outperform randomized strategies in optimality.

The method enables practical application of the Shiryaev-Roberts test with various initializations.

Abstract

For the most popular sequential change detection rules such as CUSUM, EWMA, and the Shiryaev-Roberts test, we develop integral equations and a concise numerical method to compute a number of performance metrics, including average detection delay and average time to false alarm. We pay special attention to the Shiryaev-Roberts procedure and evaluate its performance for various initialization strategies. Regarding the randomized initialization variant proposed by Pollak, known to be asymptotically optimal of order-3, we offer a means for numerically computing the quasi-stationary distribution of the Shiryaev-Roberts statistic that is the distribution of the initializing random variable, thus making this test applicable in practice. A significant side-product of our computational technique is the observation that deterministic initializations of the Shiryaev-Roberts procedure can also enjoy the same order-3 optimality property as Pollak's randomized test and, after careful selection, even uniformly outperform it.