General class of chi-square statistics for goodness-of-fit tests for stationary time series

In this contribution, a class of time-domain goodness-of-fit procedures for stationary time- series, is presented. These test procedures are based on minimum chi-square statistics in the deviations of certain sample statistics (obtained from finite-memory non-linear transformations of the process) from their ensemble counterparts. Two specific versions are derived, depending on the parameterization of the model manifold. Exact asymptotic distribution of these tests under the null hypothesis H_O and local alternatives are derived. Two applications of this general procedure is finally presented, aiming at assessing that (1) a stationary scalar time-series is autoregressive and (2) that a multivariate stationary time-series is a noisy instantaneous mixture of independent scalar time-series.