Composite likelihood estimation for a gaussian process under fixed domain asymptotics

TL;DR

This paper investigates the properties of pairwise likelihood estimators for Gaussian process covariance parameters under fixed-domain asymptotics, highlighting conditions for consistency and asymptotic normality.

Contribution

It provides new theoretical insights into the consistency and asymptotic behavior of pairwise likelihood estimators for Gaussian processes, including conditions for their effectiveness.

Findings

Weighted pairwise maximum likelihood estimator can be inconsistent depending on parameter range.

Weighted pairwise conditional maximum likelihood estimator is always consistent.

Both estimators are asymptotically Gaussian when consistent.

Abstract

We study the problem of estimating the covariance parameters of a one-dimensional Gaussian process with exponential covariance function under fixed-domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent. This depends on the range of admissible parameter values in the likelihood optimization. On the other hand, the weighted pairwise conditional maximum likelihood estimator is always consistent. Both estimators are also asymptotically Gaussian when they are consistent. Their asymptotic variances are larger or strictly larger than that of the maximum likelihood estimator. A simulation study is presented in order to compare the finite sample behavior of the pairwise likelihood estimators with their asymptotic distributions. For more general covariance functions, an additional inconsistency result is provided, for the weighted pairwise maximum likelihood estimator of a variance parameter.