Nonparametric variogram and covariogram estimation with Fourier-Bessel matrices

The nonparametric estimation of variograms and covariograms for isotropic stationary spatial stochastic processes is considered. It is shown that Fourier-Bessel matrices play an important role in this context because they provide an orthogonal discretization of the spectral representation of positive definite functions. Their properties are investigated and an example from a simulated two-dimensional spatial process is provided. It is shown that this approach provides a smooth and positive definite nonparametric estimator in the continuum, whereas previous methods typically suffer from spurious oscillations. A practical example from Astronomy is used for illustration.