Global smoothness estimation of a Gaussian process from general sequence designs

Abstract

We consider a real Gaussian process $X$ with global unknown smoothness $(r_{\scriptscriptstyle 0},\beta _{\scriptscriptstyle 0}$): more precisely $X^{(r_{\scriptscriptstyle 0})}$, $r_{\scriptscriptstyle 0}\in{\mathds {N}}_{0}$, is supposed to be locally stationary with Hölder exponent $\beta _{\scriptscriptstyle 0}$, $\beta _{\scriptscriptstyle 0}\in]0,1[$. For $X$ observed at a finite set of points, we derive estimators of $r_{\scriptscriptstyle 0}$ and $\beta _{\scriptscriptstyle 0}$ based on the quadratic variations for the divided differences of $X$. Under mild conditions, we obtain an exponential bound for estimating $r_{\scriptscriptstyle 0}$, as well as sharp rates of convergence (up to logarithmic factors) for the estimation of $\beta _{\scriptscriptstyle 0}$. An extensive simulation study illustrates the finite-sample properties of both estimators for different types of processes and we also include two real data applications.