Maximum likelihood estimation for Gaussian processes under inequality constraints

TL;DR

This paper investigates covariance parameter estimation for Gaussian processes under inequality constraints, demonstrating that constrained maximum likelihood estimators perform similarly asymptotically but often yield more accurate finite-sample results.

Contribution

It shows that constrained maximum likelihood estimators have the same asymptotic distribution as unconstrained ones and are more accurate in finite samples under inequality constraints.

Findings

Constrained MLE has the same asymptotic distribution as unconstrained MLE.

Constrained MLE is generally more accurate in finite samples.

Extensions to prediction and noisy observations are provided.

Abstract

We consider covariance parameter estimation for a Gaussian process under inequality constraints (boundedness, monotonicity or convexity) in fixed-domain asymptotics. We address the estimation of the variance parameter and the estimation of the microergodic parameter of the Mat\'ern and Wendland covariance functions. First, we show that the (unconstrained) maximum likelihood estimator has the same asymptotic distribution, unconditionally and conditionally to the fact that the Gaussian process satisfies the inequality constraints. Then, we study the recently suggested constrained maximum likelihood estimator. We show that it has the same asymptotic distribution as the (unconstrained) maximum likelihood estimator. In addition, we show in simulations that the constrained maximum likelihood estimator is generally more accurate on finite samples. Finally, we provide extensions to prediction and to noisy observations.