Detecting When a Monotonically Increasing Mean Has Crossed a Threshold

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Abstract
For monitoring a sequence of random variables, the cumulative sum (CUSUM) sequential change-point detection scheme has optimality properties if the mean experiences a single, one-time jump increase from one known level to another. However, many monitoring situations are not realistically described by this stylized change-point model. For example, in modeling tool wear, gradual monotonic changes in the mean should be allowed. In this paper, we introduce a model that assumes only that the mean is nondecreasing over time and investigate how to let the process continue as long as its mean is under some specified threshold value, stopping it as soon as possible after the mean exceeds the threshold. We show how to apply the CUSUM and the exponentially weighted moving average (EWMA) to this problem as well as compare these procedures to a repeated generalized likelihood ratio test (GLR) designed specifically for the monotone setting. A simulation study demonstrates that the CUSUM and EWMA, properly applied, perform surprisingly well compared to the GLR test, usually outperforming it. We argue that the CUSUM is the best overall choice.