2015 QPRC

ON EFFICIENT PERFORMANCE EVALUATION OF THE CUMULATIVE SUM CHART AND THE SEQUENTIAL
PROBABILITY RATIO TEST

Aleksey S. Polunchenko
Department of Mathematical Sciences
State University of New York at Binghmaton
Binghamton, New York, United States of America
eMail: aleksey@binghamton.edu
Homepage: http://www.math.binghamton.edu/aleksey

Abstract
Wald’s likelihood ratio identity is one of the fundamental tools in all of sequential analysis. This
powerful change-of-measure technique essentially enabled the proof of nearly every classical result in the field: strong optimality of Wald’s SPRT, exact minimax optimality of Page’s CUSUM chart, and exact multi-cyclic optimality of the Shiryaev–Roberts procedure—to name a few. More recently, however, Polunchenko et al. (2014a,b) put the technique to a di
erent, more applicative use: to improve the accuracy and e ciency of the numerical method they developed to compute the performance of the Shiryaev–Roberts procedure. We extend the ideas of Polunchenko et al. (2014a,b) and employ Wald’s likelihood ratio identity to establish an explicit connection between certain in-control characteristics of the CUSUM Run Length and their out-of-control counterparts. The connection is obtained in the form of paired integral (renewal) equations and the derivation exploits the fact that CUSUM is equivalent to repetitive application of the SPRT. The Run Length characteristics of interest include the whole distribution and its entire moment series (starting from the zero-state ARL). One practical benefit of the established connection is that it enables concurrent evaluation of the in- and out-of-control characteristics of the CUSUM Run Length. This is a considerable reduction of the computational burden. Moreover, due to the well-known relation between the CUSUM chart and the SPRT, the ASN and OC functions of the latter can be computed simultaneously as well. We illustrate the main ideas using CUSUM’s zero-state in- and out-of-control ARLs as an example.

References
Polunchenko, A. S., Sokolov, G., and Du, W. (2014a). An accurate method for determining the pre-change Run Length distribution of the Generalized Shiryaev–Roberts detection procedure. Sequential Analysis, 33(1):112–134.

Polunchenko, A. S., Sokolov, G., and Du, W. (2014b). E cient performance evaluation of the Generalized Shiryaev–Roberts detection procedure in a multi-cyclic setup. Applied Stochastic Models in Business and Industry, 30(6):723–739.