Abstract. The nonparametric estimation of the power spectral density of uniformly-spaced data with missing samples is revisited. Classical estimators, such as the standard periodogram and the Lomb-Scargle periodogram, are biased when samples are missing. The classical method to obtain an asymptotically-unbiased estimator is to take the finite Fourier transform of the standard unbiased estimator of the autocorrelation function. However, the latter estimator is not necessarily positive semidefinite, so its finite Fourier transform can yield negative power spectral density values at some frequencies. To avoid this problem, Gao et al. (2021) have proposed taking the absolute value of the finite Fourier transform of the standard unbiased estimator of the autocorrelation function to estimate the power spectral density of data with missing samples. We show that the estimator of power spectral density proposed by Gao et al. (2021) is even more biased than classical estimators and should not be used for quantitative analysis of spectral characteristics such as spectral slope in log-log space. We illustrate this using both synthetic data from fractional Brownian processes and actual data from a laboratory experiment of decaying turbulence in an active grid-generated air flow, to which we apply synthetic Bernoulli and batch-Bernoulli sampling functions to simulate missing samples. In fact, negative values of power spectral density estimates for particular realizations of a random process with missing samples should be retained, so that when sufficiently averaged the estimate will be nonnegative, and will not contain the bias induced from taking absolute values as Gao et al. (2021) propose. It is also proposed here to use the circular unbiased estimator of the autocorrelation function, the finite Fourier transform of which yields a power spectral density estimator identical to the standard periodogram estimator in the absence of missing samples. Its advantages are reduced variance and reduced computing memory usage compared to the finite Fourier transform of the standard unbiased estimator of the autocorrelation function. Both power spectral density estimators, when sufficiently averaged, are able to recover the -5/3 spectral slope of the decaying turbulence data even when 50 % of the data are missing. A Matlab implementation of the proposed estimator is provided.
Cédric Chavanne (Fri,) studied this question.