On the efficiency of spectral density estimates for stationary process

We consider the problem of estimating the spectral density of a stationary real-valued stochastic process in discrete time. Under a certain mixing condition we find the optimal (in the sense of asymptotic mean square error) spectrograph estimate and show that this estimate has a mean square error which is considerably less than that of usual spectro ­ graph estimates. We then study a lag time window estimate suggested by A. N. Kolmogorov and show that its mean square error is very close to the optimal one and that this estimate is less sensitive to noise and non-stationary phenomena at remote frequencies. Further, Kolmogorov's estimate turns out to be very well suited for computation by electronic computers.