Sign invariance in goodness-of-fit test for time series (Q2742776)

The authors investigated goodness-of-fit tests for stationary Gaussian processes based on a functional of the difference between the observed empirical standardized spectral distribution and the hypothesized standardized spectral distribution. Tests of this type include those based on the Kolmogorov-Smirnov and Cramér-von Mises statistics. Rather general theorems are proved to show that under certain conditions the distribution of a symmetric functional based on observations from a Gaussian process \(\{y_t(\theta)\}\) indexed by a parameter \(\theta\) is the same for \(\theta=\theta_0\) and \(\theta=-\theta_0\).NEWLINENEWLINENEWLINEThe results are illustrated by three examples of time series processes useful for applications: AR(1) models, MA(1) processes and general ARMA models.