Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process
Journal of Multivariate Analysis
The asymptotic distribution of REML estimators
Journal of Multivariate Analysis
Efficient Global Optimization of Expensive Black-Box Functions
Journal of Global Optimization
Predictive Approaches for Choosing Hyperparameters in Gaussian Processes
Neural Computation
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Toeplitz and circulant matrices: a review
Communications and Information Theory
Design and Analysis of Experiments
Design and Analysis of Experiments
Numerical Recipes 3rd Edition: The Art of Scientific Computing
Numerical Recipes 3rd Edition: The Art of Scientific Computing
Design of computer experiments: space filling and beyond
Statistics and Computing
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Calibration of computer models with multivariate output
Computational Statistics & Data Analysis
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Covariance 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 covariance parameters. The asymptotic covariance matrices of the covariance parameter estimators are deterministic functions of the regularity parameter. By means of an exhaustive study of the asymptotic covariance matrices, it is shown that the estimation is improved when the regular grid is strongly perturbed. Hence, an asymptotic confirmation is given to the commonly admitted fact that using groups of observation points with small spacing is beneficial to covariance function estimation. Finally, the prediction error, using a consistent estimator of the covariance parameters, is analyzed in detail.