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

2024/04/26 by Lucas Reding, Reding, Lucas, Andrés F. López-Lopera +3
Computer Science · Decision Sciences · Mathematics · #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Scientific Measurement and Uncertainty Evaluation #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2404.17222

openalex publication_date 2024/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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