We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors (SnIn the AR(1) case, we also give the explicit rate function for the bivariate random sequence (Wn)) n 2 , as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.
We employ a general Monte Carlo method to test composite hypotheses of goodness-of-fit for several popular multivariate models that can accommodate both asymmetry and heavy tails. Specifically, we consider weighted L2-type tests based on a discrepancy measure involving the distance between empirical characteristic functions and thus avoid the need for employing corresponding population quantities which may be unknown or complicated to work with. The only requirements of our tests are that we should be able to draw samples from the distribution under test and possess a reasonable method of estimation of the unknown distributional parameters. Monte Carlo studies are conducted to investigate the performance of the test criteria in finite samples for several families of skewed distributions. Real-data examples are also included to illustrate our method.
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