Consistent Estimation of Non-bandlimited Spectral Density From Uniformly Spaced Samples

Abstract

In the matter of selection of sample time points for the estimation of the power spectral density of a continuous time stationary stochastic process, irregular sampling schemes such as Poisson sampling are often preferred over regular (uniform) sampling. A major reason for this preference is the well-known problem of inconsistency of estimators based on regular sampling, when the underlying power spectral density is not bandlimited. It is argued in this paper that, in consideration of a large sample property like consistency, it is natural to allow the sampling rate to go to infinity as the sample size goes to infinity. Through appropriate asymptotic calculations under this scenario, it is shown that the smoothed periodogram based on regularly spaced data is a consistent estimator of the spectral density, even when the latter is not band-limited. It transpires that, under similar assumptions, the estimators based on uniformly sampled and Poisson-sampled data have about the same rate of convergence. Apart from providing this reassuring message, the paper also gives a guideline for practitioners regarding appropriate choice of the sampling rate. Theoretical calculations for large samples and Monte-Carlo simulations for small samples indicate that the smoothed periodogram based on uniformly sampled data have less variance and more bias than its counterpart based on Poisson sampled data.

Asymptotic distribution of a consistent cross-spectrum estimator based on uniformly spaced samples of a non-bandlimited process

“…It has been shown in Srivastava and Sengupta (2010) that under the assumptions of Theorems 2 and 5, the optimal rate of convergence for mean square consistency of the estimator (1) is given as…”

“…However, when one has the resources to increase the sample size indefinitely, one would like to use some of those resources to sample faster, rather than being constrained by a fixed sampling rate. In fact, it has been shown (Srivastava and Sengupta, 2010) that if this constraint is removed, and the sampling rate is allowed to increase suitably as the sample size goes to infinity, then the smoothed periodogram can be a consistent estimator of a non-bandlimited spectral density.…”

“…In Section 2, we establish the consistency of the spectrum estimator (1) for non-bandlimited processes, which is a generalization of a result of Srivastava and Sengupta (2010) to the case of multivariate time series. It paves the way for our main result on the asymptotic distribution of the estimator, given in Section 3.…”

Nonparametric estimation of the service time distribution in the M/G/∞ queue

“…Nonparametric estimation of covariance functions for continuous-time stationary processes from discrete observations is discussed in and Hall & Patil (1994). For other related work we refer to Masry (1983), Haberzettl (1997), Srivastava & Sengupta (2010) and references therein.…”

Effect of Inter-Sample Spacing Constraint on Spectrum Estimation with Irregular Sampling

“…However, consistency is only a large sample property of an estimator. Srivastava and Sengupta [6] showed that, if one has the ability to sample the process arbitrarily fast, then one can consistently estimate a non-bandlimited spectral density through uniformly spaced samples also, provided the sampling rate goes to infinity at a suitable rate as the sample size goes to infinity. By comparing the smoothed periodogram estimator with the corresponding estimator based on the Poisson process sampling, they found that, under certain regularity conditions, the rates of convergence for the two estimators are comparable and the constants associated with the rates of convergence have a trade-off in terms of bias and variance.…”