E-detectors: a nonparametric framework for online changepoint detection

Sequential changepoint detection is a classical problem with a variety of applications. However, the majority of prior work has been parametric, for example, focusing on exponential families. We develop a fundamentally new and general framework for changepoint detection when the pre- and post-change distributions are nonparametrically specified (and thus composite). Our procedures come with clean, nonasymptotic bounds on the average run length (frequency of false alarms). In certain nonparametric cases (like sub-Gaussian or sub-exponential), we also provide near-optimal bounds on the detection delay following a changepoint. The primary technical tool that we introduce is called an e-detector, which is composed of sums of e-processes -- a fundamental generalization of nonnegative supermartingales -- that are started at consecutive times. We first introduce simple Shiryaev-Roberts and CUSUM-style e-detectors, and then show how to design their mixtures in order to achieve both statistical and computational efficiency. We demonstrate their efficacy in detecting changes in the mean of a bounded random variable without any i.i.d. assumptions, with an application to tracking the performance of a sports team over multiple seasons.