Annals of Statistics

Goodness of Fit Tests for Spectral Distributions

T. W. Anderson

Full-text: Open access


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

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

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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

Export citation