Parametric estimators for stationary time series with missing observations

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Abstract
Three related estimators are considered for the parametrized spectral density of a discrete-time processX(n),n= 1, 2, · · ·, when observations are not available for all the valuesn= 1(1)N. Each of the estimators is obtained by maximizing a frequency domain approximation to a Gaussian likelihood, although they do not appear to be the most efficient estimators available because they do not fully utilize the information in the processa(n) which determines whetherX(n) is observed or missed. One estimator, called M3, assumes that the second-order properties ofa(n) are known; another, M2, lets these be known only up to an unknown parameter vector; the third, M1, requires no model fora(n). Under representative sets of conditions, which allow for both deterministic and stochastica(n), the strong consistency and asymptotic normality of M1, M2, and M3 are established. The conditions needed for consistency whenX(n) is an autoregressive moving-average process are discussed in more detail. It is also shown that in general M1 and M3 are equally efficient asymptotically and M2 is never more efficient, and may be less efficient, than M1 and M3.