A Sieve Estimator for the Covariance of a Gaussian Process

Abstract

Maximum likelihood estimation for the covariance $R$ of a zero-mean Gaussian process is considered, with no assumptions on the covariance or the "time" parameter set $T$. It is shown that the likelihood function is a.s. unbounded in general, and a sieve estimator $\hat{R}$ is constructed. The distribution of $\hat{R}$, considered as a process on $T \times T$, can be described exactly if a certain technical assumption is satisfied concerning the bivariate series expansion of $R$. It is then shown that $\hat{R}(s, t)$ is asymptotically unbiased and consistent (weakly and in mean square) at each $(s, t) \in T \times T$, and that $\hat{R}$ is strongly consistent (globally) in an appropriate norm.