Quickest Detection of Parameter Changes in Stochastic Regression: Nonparametric CUSUM

Abstract

We consider the problem of detecting abrupt parameter changes in a stochastic regression with unknown noise distribution. The process changes at some unknown point of time. Under general conditions on the regression function and unknown distributions of observations before and after the disruption, the paper develops a nonparametric cumulative sum procedure (CUSUM). Unlike likelihood-based CUSUM algorithms, constructed mostly on log-likelihood ratio statistics, we use a special system of basic statistics in Page's procedure. By applying a sequential sampling scheme, which measures time in terms of accumulated Kullback-Leibler (K-L) divergence, we come to a system of statistics with the martingale properties close to those of the log-likelihood ratios. The proposed approach suggests also an alternative performance criterion in the analysis of the procedure by replacing the expected detection delay by the corresponding K-L divergence. We show that, under the false alarm probability constraint, the nonparametric CUSUM rule is optimal in the sense that it ensures the logarithmic asymptotic for the detection delay.