Accurate estimation of the autocovariance function of random observations

Abstract

The sample autocovariance function, estimated with mean lagged products of random observations, is the Fourier transform of the raw periodogram. Therefore, the quality of the sample autocovariance as a representation of stochastic data is as poor as that of a periodogram. However, the spectral density and the autocovariance function can be estimated more accurately with a parametric method, with time series models. A recent development in time series analysis gives the possibility to select automatically the type and the order of the best time series model for data with unknown characteristics. The spectral accuracy of the selected model is better than the accuracy of all variants of periodograms. Also the accuracy of the parametric estimate of the autocovariance function is the same or better than what can be achieved by the non-parametric mean-lagged-product estimates, for every individual lag. More important, the estimated time series parameters define the autocovariance as a complete function, for all lags together.