Detecting when a monotonically increasing mean has crossed a threshold

For monitoring a sequence of random variables, the cumulative sum (CUSUM) s equential change-point detection scheme has optimality properties if the me an experiences a single, one-time jump increase from one known level to ano ther. However, many monitoring situations are not realistically described b y this stylized change-point model. For example, in modeling tool wear, gra dual monotonic changes in the mean should be allowed. In this paper. we int roduce a model that assumes only that the mean is nondecreasing over time a nd 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 exponent ially weighted moving average (EWMA) to this problem as well as compare the se procedures to a repeated generalized likelihood ratio test (GLR) designe d specifically for the monotone setting. A simulation study demonstrates th at 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.