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It is common to model a deterministic response function, such as the outputof a computer experiment, as a Gaussian process with a Mat\'ern covariancekernel. The smoothness parameter of a Mat\'ern kernel determines many importantproperties of the model in the large data limit, including the rate ofconvergence of the conditional mean to the response function. We prove that themaximum likelihood estimate of the smoothness parameter cannot asymptoticallyundersmooth 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 Sobolevsmoothness $\nu_0 > d/2$, then the smoothness parameter estimate cannot beasymptotically less than $\nu_0$. The lower bound is sharp. Additionally, weshow that maximum likelihood estimation recovers the true smoothness for aclass of compactly supported self-similar functions. For cross-validation weprove an asymptotic lower bound $\nu_0 - d/2$, which however is unlikely to besharp. The results are based on approximation theory in Sobolev spaces and somegeneral theorems that restrict the set of values that the parameter estimatorscan take.

Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation