Asymptotic behavior of the likelihood function of covariance matrices of spatial Gaussian processes

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. copyright © 2010 ralf zimmermann.