Limit Processes for Spectral Distribution Functions with Applications to Goodness-Of-Fit Testing

摘要

The limits in distribution of the sequences of stochastic processes defined by the deviations of the sample co-spectral and quadrature spectral distribution functions are found using the theory of convergence in distribution of stochastic processes. Our study requires that the limiting central moments be found and, since we carry out our derivations entirely in the spectral domain, we require a discussion of certain kernels that arise in the computation of these moments. The limit processes are shown to be Gaussian with independent increments and with covariances defined in terms of the hypothesized spectral densities. The distributions of certain functionals on these limit processes are computed. The functionals include Cramer-Von Mises, Watson and Kuiper-type statistics as well as the Smirnov and Kolmogorov-type statistics. These results are used to obtain the limiting distribution of goodness-of-fit tests of the types mentioned avove for the co-spectral and quadrature spectral distribution functions. Normalized spectral distribution functions are also considered. (Author)