Journal of Applied Mathematics
Volume 2010 (2010), Article ID 494070, 17 pages
Research Article

Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes

German Aerospace Center (DLR), Lilienthalplatz 7, 38108 Braunschweig, Germany

Received 16 April 2010; Revised 3 November 2010; Accepted 28 November 2010

Academic Editor: F. Marcellan

Copyright © 2010 Ralf Zimmermann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The covariance structure of spatial Gaussian predictors (aka Kriging predictors) is generally modeled by parameterized covariance functions; the associated hyperparameters in turn are estimated via the method of maximum likelihood. In this work, the asymptotic behavior of the maximum likelihood of spatial Gaussian predictor models as a function of its hyperparameters is investigated theoretically. Asymptotic sandwich bounds for the maximum likelihood function in terms of the condition number of the associated covariance matrix are established. As a consequence, the main result is obtained: optimally trained nondegenerate spatial Gaussian processes cannot feature arbitrary ill-conditioned correlation matrices. The implication of this theorem on Kriging hyperparameter optimization is exposed. A nonartificial example is presented, where maximum likelihood-based Kriging model training is necessarily bound to fail.