Sign Invariance in Goodness-of-Fit Tests for Time Series

A goodness-of-fit test for a stationary stochastic process may be based on a functional of the difference between the sample standardized spectral distribution and a hypothesized standardized spectral distribution. Theorems are given to show that under certain conditions the distribution of such a functional based on observations from a process { y_t } indexed by a parameter θ is the same for θ=θ₀ and for θ=−θ₀. The results are illustrated by three examples of time series processes.