Cross-validation estimation of covariance parameters under fixed-domain asymptotics

TL;DR

This paper proves the strong consistency and asymptotic normality of a cross validation estimator for the covariance parameter of a Gaussian process under fixed-domain asymptotics, extending prior work on maximum likelihood estimation.

Contribution

It provides the first fixed-domain asymptotic analysis of cross validation for covariance parameter estimation, including variance bounds and examples of observation point arrangements.

Findings

Cross validation estimator is consistent and asymptotically normal.

Asymptotic variance of cross validation depends on observation design.

Bounds on the asymptotic variance are established and achieved by specific designs.

Abstract

We consider a one-dimensional Gaussian process having exponential covariance function. Under fixed-domain asymptotics, we prove the strong consistency and asymptotic normality of a cross validation estimator of the microergodic covariance parameter. In this setting, Ying [40] proved the same asymptotic properties for the maximum likelihood estimator. Our proof includes several original or more involved components, compared to that of Ying. Also, while the asymptotic variance of maximum likelihood does not depend on the triangular array of observation points under consideration, that of cross validation does, and is shown to be lower and upper bounded. The lower bound coincides with the asymptotic variance of maximum likelihood. We provide examples of triangular arrays of observation points achieving the lower and upper bounds. We illustrate our asymptotic results with simulations, and provide extensions to the case of an unknown mean function. To our knowledge, this work constitutes the first fixed-domain asymptotic analysis of cross validation.