Cumulative Sum Algorithms Based on Nonparametric Kernel Density Estimators

Cumulative sum (CUSUM) algorithms are used for monitoring in various applications, including manufacturing, network monitoring, financial markets, biosurveillance, and many more. A popular CUSUM technique for detecting a change in the in-control distribution of an independentdata sequence is based on repeated use of the sequential probability ratio test (SPRT). Some optimality results have been derived for the SPRT-based CUSUM when the in-control and out-of-control distributions are fully known. We introduce an approximation formula for thethreshold value of an SPRT-based CUSUM. Limited research has been performed on CUSUM techniques when the distributions are not fully specified. This research is concerned about how to use the CUSUM when the underlying in-control distribution is arbitrary and unknown, and the out-of-control density is either an additive or a multiplicative transformation of the in-control density. The proposed solution combines an adaptive nonparametric kernel density estimator derived from an in-control sample of observations with a smoothed bootstrap algorithm that enables the CUSUM to work effectively for reasonably sized sets of in-control data