On robustness of the Shiryaev–Roberts change-point detection procedure under parameter misspecification in the post-change distribution

ABSTRACT

The gist of the quickest change-point detection problem is to detect the presence of a change in the statistical behavior of a series of sequentially made observations, and do so in an optimal detection-speed-versus-“false-positive”-risk manner. When optimality is understood either in the generalized Bayesian sense or as defined in Shiryaev's multi-cyclic setup, the so-called Shiryaev–Roberts (SR) detection procedure is known to be the “best one can do”, provided, however, that the observations’ pre- and post-change distributions are both fully specified. We consider a more realistic setup, viz. one where the post-change distribution is assumed known only up to a parameter, so that the latter may be misspecified. The question of interest is the sensitivity (or robustness) of the otherwise “best” SR procedure with respect to a possible misspecification of the post-change distribution parameter. To answer this question, we provide a case study where, in a specific Gaussian scenario, we allow the SR procedure to be “out of tune” in the way of the post-change distribution parameter, and numerically assess the effect of the “mistuning” on Shiryaev's (multi-cyclic) Stationary Average Detection Delay delivered by the SR procedure. The comprehensive quantitative robustness characterization of the SR procedure obtained in the study can be used to develop the respective theory as well as to provide a rational for practical design of the SR procedure. The overall qualitative conclusion of the study is an expected one: the SR procedure is less (more) robust for less (more) contrast changes and for lower (higher) levels of the false alarm risk.

KEYWORDS:

• Sequential analysis
• Shiryaev–Roberts procedure
• Quality control
• Quickest change-point detection

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62L10, 62L15
• Secondary 62P30, 65R20

Acknowledgments

The authors are grateful to Dr. Emmanuel Yashchin of the Mathematical Sciences Department at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA; to Prof. William H. Woodall of the Statistics Department at the Virginia Polytechnic Institute (Virginia Tech), Blacksburg, Virginia, USA; to Prof. Sven Knoth of the Department of Mathematics and Statistics at the Helmut Schmidt University, Hamburg, Germany; to Dean Neubauer of Corning Incorporated, Corning, New York, USA; to Prof. Subha Chakraborti of the Department of Information Systems, Statistics and Management Science at the University of Alabama, Alabama, USA; and to Dr. Ron Kenett of Israel-based KPA Ltd. (www.kpa-group.com), for the interest in this work and for the constructive feedback that helped improve the quality of the manuscript. Additional thanks goes out to the anonymous referee for the comments and suggestions that helped to ameliorate the article further.

The effort of A.S. Polunchenko was supported, in part, by the Simons Foundation (www.simonsfoundation.org) via a Collaboration Grant in Mathematics (Award # 304574) and by the Research Foundation for the State University of New York at Binghamton via an Interdisciplinary Collaboration Grant (Award # 66761).

Last but not least, A. S. Polunchenko is also personally thankful to the Office of the Dean of the Harpur College of Arts and Sciences at the State University of New York (SUNY) at Binghamton for the support provided through the Dean's Research Semester Award for Junior Faculty granted for the Fall semester of 2014. The Award allowed to focus on this research more fully.