## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 1 (1993), 552-575.

### The Effect of Serial Correlation on Tests for Parameter Change at Unknown Time

#### Abstract

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 the tests when high frequency mass predominates. These effects are characterized by the expression ${2\pi f(0)/\int^\pi_-\pi f(\lambda)d\lambda}$, where $f(\centerdot)$ is the spectral density of the noise process; in sample survey work this is know 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.

#### Article information

**Source**

Ann. Statist. Volume 21, Number 1 (1993), 552-575.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176349042

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aos/1176349042

**Mathematical Reviews number (MathSciNet)**

MR1212193

**Zentralblatt MATH identifier**

0771.62072

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 62E20: Asymptotic distribution theory 62G10: Hypothesis testing 62J05: Linear regression 62M15: Spectral analysis

**Keywords**

Serial correlation cumulative sums change-point statistics spectral density residuals processes partial sums

#### Citation

Tang, S. M.; MacNeill, I. B. The Effect of Serial Correlation on Tests for Parameter Change at Unknown Time. Ann. Statist. 21 (1993), no. 1, 552--575. doi:10.1214/aos/1176349042. http://projecteuclid.org/euclid.aos/1176349042.