Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel
This research provides a complete asymptotic characterization for estimating Gaussian Process parameters—spatial variance ($\sigma^2$), lengthscale ($l$), and nugget variance ($\tau^2$)—using Maximum Likelihood Estimation under fixed-domain asymptotics. The results show that $\sigma^2$ and $l$ converge at slow logarithmic rates, while $\tau^2$ converges at the standard parametric rate. These rates are proven to be minimax optimal, offering a sharp understanding of estimation accuracy.