An asymptotic study of the joint maximum likelihood estimation of the regularity and the amplitude parameters of a Mat{é}rn model on the circle

This work considers parameter estimation for Gaussian process interpolation with a periodized version of the Mat{\é}rn covariance function introduced by Stein. Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data are sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that a joint estimation does not select the regularity parameter as if the amplitude were fixed.