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

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.

💡 Research Summary

The paper presents a unified numerical framework for evaluating the performance of three of the most widely used sequential change‑point detection procedures: the Cumulative Sum (CUSUM) chart, the Exponentially Weighted Moving Average (EWMA) scheme, and the Shiryaev‑Roberts (SR) test. After briefly reviewing the classical optimality criteria—minimizing the average detection delay (ADD) subject to a constraint on the average run length (ARL) to false alarm—the authors derive a pair of linear integral equations that exactly characterize the ADD and ARL for any given detection rule when the detection statistic evolves as a Markov process. The kernel of these equations is the one‑step transition density of the statistic under the post‑change hypothesis, and the boundary condition corresponds to absorption at the detection threshold.

For CUSUM and EWMA, the integral equations reduce to well‑known forms, but the authors show that the same formalism can be applied to the SR statistic, whose performance is highly sensitive to the choice of the initial value R₀. Traditionally, two initializations have been considered: the “classical” SR with R₀ = 0 and the randomized SR proposed by Pollak, in which R₀ is drawn from the quasi‑stationary distribution of the SR statistic. Pollak’s version is known to be asymptotically optimal of order three (i.e., its ADD differs from the optimal Lorden bound by a term of order o(1) as ARL → ∞). However, the quasi‑stationary distribution has never been computed in closed form, limiting practical implementation.

The authors address this gap by formulating a fixed‑point equation for the quasi‑stationary density and solving it numerically using a discretization of the state space combined with a simple iteration scheme. The resulting algorithm converges rapidly and yields the exact distribution of the random initializer required for Pollak’s test, making the third‑order optimality claim operational.

Beyond the randomized scheme, the paper investigates deterministic initializations. By scanning a grid of possible initial values r ∈