Asymptotic analysis for covariance parameter estimation of Gaussian processes with functional inputs

TL;DR

This paper establishes the asymptotic properties of covariance parameter estimators for Gaussian processes with functional inputs, including robustness to approximation errors, supported by analytical and numerical examples.

Contribution

It extends asymptotic analysis of Gaussian process covariance estimation to scenarios with input approximation errors, ensuring robustness in practical settings.

Findings

Maximum likelihood estimator is consistent and asymptotically normal.

Robustness to input approximation errors is theoretically validated.

Analytical and numerical examples illustrate the theoretical results.

Abstract

We consider covariance parameter estimation for Gaussian processes with functional inputs. From an increasing-domain asymptotics perspective, we prove the asymptotic consistency and normality of the maximum likelihood estimator. We extend these theoretical guarantees to encompass scenarios accounting for approximation errors in the inputs, which allows robustness of practical implementations relying on conventional sampling methods or projections onto a functional basis. Loosely speaking, both consistency and normality hold when the approximation error becomes negligible, a condition that is often achieved as the number of samples or basis functions becomes large. These later asymptotic properties are illustrated through analytical examples, including one that covers the case of non-randomly perturbed grids, as well as several numerical illustrations.