GOODNESS-OF-FIT TESTS FOR SPECTRAL DISTRIBUTIONS

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