Asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case

TL;DR

This paper analyzes the asymptotic behavior of covariance parameter estimators for Gaussian processes in misspecified models, showing that ML and CV estimators target different optimality criteria under irregular sampling.

Contribution

It provides a comprehensive asymptotic analysis of ML and CV estimators in the misspecified case, revealing their distinct asymptotic targets and introducing new increasing-domain results.

Findings

ML asymptotically minimizes Kullback-Leibler divergence.

CV asymptotically minimizes integrated square prediction error.

Simulations show contrasting performance of ML and CV estimators.

Abstract

In parametric estimation of covariance function of Gaussian processes, it is often the case that the true covariance function does not belong to the parametric set used for estimation. This situation is called the misspecified case. In this case, it has been shown that, for irregular spatial sampling of observation points, Cross Validation can yield smaller prediction errors than Maximum Likelihood. Motivated by this observation, we provide a general asymptotic analysis of the misspecified case, for independent and uniformly distributed observation points. We prove that the Maximum Likelihood estimator asymptotically minimizes a Kullback-Leibler divergence, within the misspecified parametric set, while Cross Validation asymptotically minimizes the integrated square prediction error. In a Monte Carlo simulation, we show that the covariance parameters estimated by Maximum Likelihood and Cross Validation, and the corresponding Kullback-Leibler divergences and integrated square prediction errors, can be strongly contrasting. On a more technical level, we provide new increasing-domain asymptotic results for independent and uniformly distributed observation points.