Goodness of Fit Tests for Spectral Distributions

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.