On the discretization of nonparametric isotropic covariogram estimators

In this article, we describe the discretization of nonparametric covariogram estimators for isotropic stationary stochastic processes. The use of nonparametric estimators is important to avoid the difficulties in selecting a parametric model. The key property the isotropic covariogram must satisfy is to be positive definite and thus have the form characterized by Yaglom's representation of Bochner's theorem. We present an optimal discretization of the latter in the sense that the resulting nonparametric covariogram estimators are guaranteed to be smooth and positive definite in the continuum. This provides an answer to an issue raised by Hall, Fisher and Hoffmann (1994). Furthermore, from a practical viewpoint, our result is important because a nonlinear constrained algorithm can sometimes be avoided and the solution can be found by least squares. Some numerical results are presented for illustration.