Asymptotic Theory for Kriging with Estimated Parameters and Its Application to Network Design (preliminary Version)

A second-order expansion is established for predictive distributions in Gaussian processes with estimated covariances. Particular focus is on estimating quantiles of the predictive distribution and their subsequent application to prediction intervals. Two basic approaches are considered, (a) a “plug-in” approach using the restricted maximum likelihood estimate of the covariance parameters, (b) a Bayesian approach using general priors. Calculations of “coverage probability bias” show that the Bayesian approach is superior in the tails of the predictive distributions, regardless of the prior. However they also imply the existence of a “matching prior” for which the second-order coverage probability bias vanishes. Previously suggested frequentist corrections do not have this property, but we use our results to suggest a new frequentist approach that does. We also compute the expected length of a Bayesian prediction interval, suggesting that this might be used as a design criterion combining recent “estimative” and “predictive” approaches to network design. A surprising parallel emerges with the recent two-stage estimative-predictive approach of Zhu and Stein.