Similarity law in spectral estimation of a time series. IV

A continuation of V. Yu. Terebizh, Astrofizika,40, 139, 273, 413 (1997). An explicit representation is found for the Fisher matrix for spectral density, enabling one to calculate the lower limit of the variance of an arbitrary unbiased density estimate. Basic equations describing smoothed density estimates are given for comparison with exact results. By numerical modeling based on the example of an AR-1 process, it is shown that the relative accuracy q of estimation of density is some universal function of the parameter w = (F -1)/N, where F is the number of parameters underlying the estimate and N is the length of the time series. The relationship q = θ(w), a similarity law, explains why a number of density estimates proposed earlier (Schuster’s periodogram, in particular) proved to be statistically inconsistent. It is just these estimates that presume an extremely detailed model of spectral density. The need for the complexity of the model to be consistent with the observational data follows from the limitation of information about the spectrum of the random process included in a sample of readings from a series of fixed volume. Translated from Astrofizika, Vol. 41, No. 1. pp. 113–122, January-March, 1998.