Global smoothness estimation of a Gaussian process from general sequence designs

Abstract

We consider a real Gaussian process X with global unknown smoothness (γ0, β0):more precisely X(γ,0) γ0 ∈ N0is supposed to be locally stationary with Hölder exponent β0, β0 γ]0,1[. For X observed at a finite set of points, we derive estimators of r0and β0based on the quadratic variations for the divided differences of X. Under mild conditions, we obtain an exponential bound for estimating β0, as well as sharp rates of convergence (up to logarithmic factors) for the estimation of β0. 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.