Asymptotic Confidence Intervals of Spectra and Cross-spectra of Continuous Time Processes Based on Uniformly Spaced Samples

It is well known that if the power spectrum of a continuous time stationary stochastic process is not limited to a finite interval, data sampled from that process at any uniform sampling rate leads to biased and inconsistent spectrum estimators. As a result, depending on the range of frequency of interest, one uses anti-alias filtering and subsequently treats the filtered process as bandlimited. However, if the spectrum has a large peak in the ‘stop band’ of the filter, and the filter is not an ideal one, then spectrum estimation would be problematic, even if the sample size is large. In this paper, we use the well-known smoothed periodogram estimator to construct asymptotic confidence intervals shrinking to the true spectra, by taking into account the decay rate of the filtered process and allowing the sampling rate to go to infinity suitably fast as the sample size goes to infinity. The proposed method requires minimal computation, as it does not involve bootstrap or other resampling. The method is illustrated through a MonteCarlo simulation study, and its performance is compared with conventional methods based on uniform and Poisson sampling.