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.

翻译：通常将确定性响应函数（例如计算机实验的输出）建模为具有Matérn协方差核的高斯过程。Matérn核的光滑参数决定了模型在大数据极限下的许多重要性质，包括条件均值向响应函数的收敛速度。我们证明，当数据在$\mathbb{R}^d$的固定有界子集上获得时，光滑参数的最大似然估计无法渐近地低估真实值。即，如果生成数据的响应函数具有Sobolev光滑度$\nu_0 > d/2$，则光滑参数估计不能渐近地小于$\nu_0$。该下界是紧的。此外，我们表明，对于一类紧支撑自相似函数，最大似然估计能恢复真实光滑度。对于交叉验证，我们证明了一个渐近下界$\nu_0 - d/2$，但该下界可能不紧。这些结果基于Sobolev空间中的逼近理论以及一些限制参数估计量可取值的通用定理。