The Whittle estimator for strongly dependent stationary Gaussian fields

In this article we generalize results on the asymptotic behaviour of the Whittle estimator for certain stationary Gaussian long range dependent fields. These results have been established in the one-dimensional case under very general conditions. They require controlling the estimation bias and also giving convergence theorems for certain quadratic forms of the observations. In the multidimensional setting, our main interest will be controlling the bias. This can be done for d ≤ 3 using taper functions, and, depending on the shape of the singularity, also introducing certain regularizing functions. In this last case, however, the estimator will no longer be efficient. We also present certain partial results concerning the convergence to a limiting Gaussian distribution of the associated quadratic forms.