The estimation of the covariances of a stationary time series with missing observations is considered. General formulas for the asymptotic second-order moments of the sample covariances of such a time series are given for either random or deterministic patterns of misses. Closed-form expressions are derived for the random Bernoulli pattern and for the deterministic periodic pattern of missing observations and are explicitly evaluated for autoregressive moving-average time series. These results are useful for constructing and analyzing parameter or spectrum estimation algorithms based on the sample covariances for a stationary time series with missing observations.>

The problem of spectral estimation through the autoregressive moving-average (ARMA) modeling of stationary processes with missing observations is considered. A class of estimators based on the sample covariances is presented, and an asymptotically optimal estimator in this class is proposed. The proposed algorithm is based on a nonlinear-least-squares fit of the sample covariances computed from the data to the true covariances of the assumed ARMA model. The statistical properties of the algorithm are explored and used to show that it is asymptotically optimal, in the sense of achieving the smallest possible asymptotic variance. The performance of the algorithm is illustrated by some numerical examples.>

In this paper we consider the problem of spectral estimation through ARMA modeling of stationary time series witch missing observations. We consider estimators based on the sample covariances, and present an asymptotically optimal estimator among this group. The algorithm is based on a nonlinear least squares fit of the sample covariances computed from the data with missing observations to the true covariances of the assumed ARMA model. The statistical properties of the algorithm are shown to be asymptotically optimal. The performance of the algorithm is illustrated by a numerical example.

An asymptotic formula is derived for the Cramer-Rao lower bound on unbiased estimates of the parameters of Gaussian autoregressive moving-average (ARMA) processes, in the case where the measurements are not contiguous, but follow a periodic pattern of misses. The formula is then used to illustrate the behavior of the bound for some specific examples.