Covariance parameter estimation of Gaussian processes with approximated functional inputs

We consider the problem of covariance parameter estimation for Gaussian processes with functional inputs. Our study addresses scenarios where exact functional inputs are available and where only approximate versions of these functions are accessible. From an increasing-domain asymptotics perspective, we first establish the asymptotic consistency and normality of the maximum likelihood estimator for the exact inputs. Then, by accounting for approximation errors, we certify the robustness of practical implementations that rely on conventional sampling methods or projections onto a functional basis. Loosely speaking, both consistency and normality continue to hold when the approximation error becomes negligible, a condition often met as the number of samples or basis functions becomes large. To ensure broad applicability, our asymptotic analysis is conducted for any Hilbert space of inputs. Our findings are illustrated through analytical examples, including the case of non-randomly perturbed grids, as well as several numerical illustrations.