Global smoothness estimation of a Gaussian process from regular sequence designs

Abstract

We consider a real Gaussian process $X$ having a global unknown smoothness $(r_{\scriptscriptstyle 0},β_{\scriptscriptstyle 0})$, $r_{\scriptscriptstyle 0}\in \mathds{N}_0$ and $β_{\scriptscriptstyle 0} \in]0,1[$, with $X^{(r_{\scriptscriptstyle 0})}$ (the mean-square derivative of $X$ if $r_{\scriptscriptstyle 0}\ge 1$) supposed to be locally stationary with index $β_{\scriptscriptstyle 0}$. From the behavior of quadratic variations built on divided differences of $X$, we derive an estimator of $(r_{\scriptscriptstyle 0},β_{\scriptscriptstyle 0})$ 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.