Mean shift testing in correlated data

Michael W. Robbins · Colin Gallagher · Robert Lund · Alexander Aue

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Advanced Statistical Process Monitoring Fault Detection and Control Systems Statistical and numerical algorithms

Abstract

Several tests for detecting mean shifts at an unknown time in stationary time series have been proposed, including cumulative sum (CUSUM), Gaussian likelihood ratio (LR), maximum of F(Fmax) and extreme value statistics. This article reviews these tests, connects them with theoretical results, and compares their finite sample performance via simulation. We propose an adjusted CUSUM statistic which is closely related to the LR test and which links all tests. We find that tests based on CUSUMing estimated one-step-ahead prediction residuals from a fitted autoregressive moving average perform well in general and that the LR and Fmax tests (which induce substantial computational complexities) offer only a slight increase in power over the adjusted CUSUM test. We also conclude that CUSUM procedures work slightly better when the changepoint time is located near the centre of the data, but the adjusted CUSUM methods are preferable when the changepoint lies closer to the beginning or end of the data record. Finally, an application is presented to demonstrate the importance of the choice of method.

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References

1954CONTINUOUS INSPECTION SCHEMES 1991Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation 1997Limit theorems in change-point analysis 1955A Test for a Change in a Parameter Occurring at an Unknown Point 1995Testing for a Change in the Parameter Values and Order of an Autoregressive Model 198820 Nonparametric methods for changepoint problems 1994Weak Convergence of the Sequential Empirical Processes of Residuals in ARMA Models 1974Tests for Change of Parameter at Unknown Times and Distributions of Some Related Functionals on Brownian Motion

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