Global smoothness estimation of a Gaussian process from regular sequence designs

Delphine Blanke (1) , Céline Vial (2)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 PSPM - Probabilités, statistique, physique mathématique

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

Mots clés: Inference for Gaussian processes; Locally stationary process; Divided differences; Quadratic variations

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Delphine Blanke, Céline Vial. Global smoothness estimation of a Gaussian process from regular sequence designs. 2014. ⟨hal-00750409v2⟩