Asymptotic statistics of AR spectral estimators for processes containing mixed spectrum

We address the influence of point spectrum on the large sample statistics of the AR(n) spectral estimator for fixed n as well as for the case where n approaches infinity. For fixed n we obtain the distribution of this estimator. We also obtain approximate expressions for its mean and variance. These expressions involve the nth order Capon spectrum. Using convergence properties of this spectrum as n approaches infinity, we show that these expressions depend on the ratio of the AR(n) to the nth order Capon spectrum. This ratio gives insight into the statistical influence of the point spectrum on the AR(n) spectral estimator, based on the well known difference in the resolving properties of these two spectra. Simulations are included to support the theoretical results. Finally, it is hoped that our attempt to bring to bear a number of published results in this area will also contribute to a better understanding of it, and possibly stimulate further investigations