  It is common to model a deterministic response function, such as the output
of a computer experiment, as a Gaussian process with a Mat\'ern covariance
kernel. The smoothness parameter of a Mat\'ern kernel determines many important
properties of the model in the large data limit, including the rate of
convergence of the conditional mean to the response function. We prove that the
maximum likelihood estimate of the smoothness parameter cannot asymptotically
undersmooth the truth when the data are obtained on a fixed bounded subset of
$\mathbb{R}^d$. That is, if the data-generating response function has Sobolev
smoothness $\nu_0 > d/2$, then the smoothness parameter estimate cannot be
asymptotically less than $\nu_0$. The lower bound is sharp. Additionally, we
show that maximum likelihood estimation recovers the true smoothness for a
class of compactly supported self-similar functions. For cross-validation we
prove an asymptotic lower bound $\nu_0 - d/2$, which however is unlikely to be
sharp. The results are based on approximation theory in Sobolev spaces and some
general theorems that restrict the set of values that the parameter estimators
can take.
