Moderate deviations of empirical periodogram and non-linear functionals of moving average processes.
Résumé
A moderate deviation principle for nonlinear functionals, with at most quadratic growth, of moving average processes (or linear processes) is established. The main assumptions on the moving average process are a logarithmic Sobolev inequality for the driving random variables and the continuity, or some (weaker) integrability condition on the spectral density (covering some cases of long-range dependence). We also obtain the moderate deviation estimate for the empirical periodogram, exhibiting an interesting new form of the rate function, i.e. with a correction term compared to the Gaussian rate functional. As statistical applications we provide the moderate deviation estimates of the least-square and the Yule-Walker estimators of the parameter of a stationary autoregressive process and of the Neyman-Pearson likelihood ratio test in the Gaussian case.