  This work considers parameter estimation for Gaussian process interpolation
with a periodized version of the Mat{\'e}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.
