Bias-free estimation of the covariance function and the power spectral density from data with missing samples including extended data gaps

Abstract

Nonparametric estimation of the covariance function and the power spectral density of uniformly spaced data from stationary stochastic processes with missing samples is investigated. Several common methods are tested for their systematic and random errors under the condition of variations in the distribution of the missing samples. In addition to random and independent outliers, the influence of longer and hence correlated data gaps on the performance of the various estimators is also investigated. The aim is to construct a bias-free estimation routine for the covariance function and the power spectral density from stationary stochastic processes under the condition of missing samples with an optimum use of the available information in terms of low estimation variance and mean square error, and that independent of the spectral composition of the data gaps. The proposed procedure is a combination of three methods that allow bias-free estimation of the desired statistical functions with efficient use of the available information: weighted averaging over valid samples, derivation of the covariance estimate for the entire data set and restriction of the domain of the covariance function in a post-processing step, and appropriate correction of the covariance estimate after removal of the estimated mean value. The procedures abstain from interpolation of missing samples as well as block subdivision. Spectral estimates are obtained from covariance functions and vice versa using Wiener–Khinchin’s theorem.