Design and Comparison of Shiryaev-Roberts and CUSUM-Type Change-Point Detection Procedures

Abstract. We address a simple changepoint detection problem where observations are i.i.d. before and after the change with known pre- and post-change distributions. For this setting, the CUSUM test is known to be optimal in the minimax setting for Lorden’s essential supremum metric, whereas the Shiryaev-Roberts procedure is optimal for detecting a change that occurs at a distant time horizon. At the same time, a randomized extension of the Shiryaev-Roberts test proposed by Pollak, enjoys a very strong asymptotic minimax property with respect to Pollak’s supremum metric. We conjecture that a deterministically initialized version of the Shiryaev-Roberts test can compete with the latter procedure very efficiently. We propose a numerical scheme for the systematic comparison of these detection procedures in both settings, i.e., minimax and for detecting changes that occur in the distant future. Our goal is accomplished by deriving a set of integral equations for the performance metrics of interest, which are solved numerically. We present numerical results for the problem of detecting a change in the mean of an exponential sequence which justify our conjecture and allow for a precise comparison of a number of changepoint detection procedures