## The effect of serial correlation on tests for parameter change at unknown time.(English)Zbl 0771.62072

Summary: It is shown that serial correlation can produce striking effects in distributions of change-point statistics. Failure to account for these effects is shown to invalidate change-point tests, either through increases in the type 1 error rates if low frequency spectral mass predominates in the spectrum of the noise process, or through diminution of the power of tests when high frequency mass predominates. These effects are characterized by the expression $$\bigl\{2\pi f(0)/\int^ \pi_{-\pi} f(\lambda)d\lambda\bigr\}$$, where $$f(\cdot)$$ is the spectral density of the noise process; in sample survey work this is known as the design effect or “deff”.
Simple precise adjustments to change-point test statistics which account for serial correlation are provided. The same adjustment applies to all commonly used regression models. Residual processes are derived for both stationary time series satisfying a moment condition and for general linear regression models with stationary error structure.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 62G10 Nonparametric hypothesis testing 62J05 Linear regression; mixed models 62M15 Inference from stochastic processes and spectral analysis
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