Asymptotic analysis of the role of spatial sampling for hyper-parameter estimation of Gaussian processes

Hyper-parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar regularity parameter. Consistency and asymptotic normality are proved for the Maximum Likelihood and Cross Validation estimators of the hyper-parameters. The asymptotic covariance matrices of the hyper-parameter estimators are deterministic functions of the regularity parameter. By means of an exhaustive study of the asymptotic covariance matrices, it is shown that irregular sampling is generally an advantage to estimation, but we identify cases where it is not the case. Therefore, a negative answer is given to the claim that irregular sampling is always better for hyper-parameter estimation than regular sampling.