Global smoothness estimation of a Gaussian process from regular sequence designs

We consider a real Gaussian process $X$ having a global unknown smoothness $(r_0,{\beta }_0)$, $r_0\in \text{mathds}{N}_0$ and $\beta_{\scriptscriptstyle 0} \in]0,1[$,with$X^{(r_{\scriptscriptstyle 0})}$(themean-squarederivativeof$X$if$r_{\scriptscriptstyle 0}\ge 1$) supposed to be locally stationary with index ${\beta }_0$. From the behavior of quadratic variations built on divided differences of $X$, we derive an estimator of $(r_0,{\beta }_0)$ based on - not necessarily equally spaced - observations of $X$. Various numerical studies of these estimators exhibit their properties for finite sample size and different types of processes, and are also completed by two examples of application to real data.