An Estimating Method for Parametric Spectral Densities of Gaussian Time Series

An estimating method for spectral densities of Gaussian time series that belong to a parametric model is proposed. Spectral density estimators are evaluated by using average Kullback–Leibler divergence from the true spectral density to estimated spectral densities. In the classical approach, unknown spectral densities are estimated by replacing the unknown parameters by asymptotically efficient estimates. In the alternative method introduced in the present paper, spectral density estimates usually do not belong to the model. The alternative spectral density estimators asymptotically dominate the classical ones. The difference in average Kullback–Leibler divergence between them can be regarded as the mixture mean curvature of the model in the space of all spectral densities. The explicit expression for the proposed estimators of spectral densities of autoregressive processes is obtained. The accuracy of prediction can be improved by using predictors that correspond to the alternative spectral density estimators.