Tests for Change in a Mean Function when the Data are Dependent

Abstract

Detecting changes in the mean of a stochastic process is important in many areas, including quality control. We develop powerful omnibus tests for the null hypothesis that the underlying mean is constant. The proposed tests can be applied to test for any kind of change, whether it be abrupt, smooth or cyclical. Nonparametric function estimation techniques are used in deriving these tests. The test statistics are derived from a Fourier series smoother that minimizes an estimate of mean integrated squared error. An important example of correlated data is that arising from a stationary time series. To obtain a valid test of mean constancy, it is necessary to estimate the spectrum of the error process, and we consider various methods of doing this. We have found that, in the case of an AR(1) model, the spectrum is well estimated if local linear smoothing is used in conjunction with generalized least squares. A power study of the proposed tests is done by simulation when the errors follow an AR(1) model, and the tests are applied to a set of astronomy data.

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