Detecting When a Monotonically Increasing Mean Has Crossed a Threshold

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.

• Change Point
• Cumulative Sum
• Exponentially Weighted Moving Average
• Generalized Likelihood Ratio
• Monte Carlo Simulation
• Regression Analysis
• Statistical Process Control

Additional information

Notes on contributors

Joseph T. Chang

Dr. Chang is an Associate Professor in the Statistics Department. His email address is [email protected].

Ronald D. Fricker

Dr. Fricker is an Associate Statistician. He is a Member of ASQ.