Modified Cramer-von Mises Goodness-of-Fit Criterion for Time Series

摘要

In this paper we consider somewhat analogous quadratic forms in normal variableswhen the dimensionality is infinite. Then the quadratic forms are distributed infinite weighted SUMS Of X2 -variables. These come about as goodness-of-fit criteria for a hypothesis that a cumulative distribution function is a specified one or that two cdf's are the same. Such criteria also arise for goodness-of-fit tests for standardized spectral distributions. As examples, we give tables of the distribution of the criterion for testing the hypothesis that a stationary stochastic process is a given moving average process order 1 and for testing the hypothesis that it is a specified autoregressive process order 1. Two methods are described for calculating the distribution. Either method is appropriate for calculating the distribution of the criterion for testing the hypothesis that a process is a stationary process whose standardized spectral density of distribution is a specified one. Goodness of fit, Time series, Mahalanobis distance, Stationary stochastic process, Spectral distributions.