Annals of Statistics

Goodness of Fit Tests for Spectral Distributions

T. W. Anderson

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Abstract

The spectral distribution function of a stationary stochastic process standardized by dividing by the variance of the process is a linear function of the autocorrelations. The integral of the sample standardized spectral density (periodogram) is a similar linear function of the autocorrelations. As the sample size increases, the difference of these two functions multiplied by the square root of the sample size converges weakly to a Gaussian stochastic process with a continuous time parameter. A monotonic transformation of this parameter yields a Brownian bridge plus an independent random term. The distributions of functionals of this process are the limiting distributions of goodness of fit criteria that are used for testing hypotheses about the process autocorrelations. An application is to tests of independence (flat spectrum). The characteristic function of the Cramer-von Mises statistic is obtained; inequalities for the Kolmogorov-Smirnov criterion are given. Confidence regions for unspecified process distributions are found.

Article information

Source
Ann. Statist., Volume 21, Number 2 (1993), 830-847.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349153

Digital Object Identifier
doi:10.1214/aos/1176349153

Mathematical Reviews number (MathSciNet)
MR1232521

Zentralblatt MATH identifier
0779.62083

JSTOR
links.jstor.org

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M15: Spectral analysis

Keywords
Goodness of fit tests spectral distributions Cramer-von Mises test Kolmogorov-Smirnov test Fredholm determinant

Citation

Anderson, T. W. Goodness of Fit Tests for Spectral Distributions. Ann. Statist. 21 (1993), no. 2, 830--847. doi:10.1214/aos/1176349153. https://projecteuclid.org/euclid.aos/1176349153


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