Assessing the number of mean-square derivatives of a Gaussian process
Résumé
We consider a real Gaussian process $X$ with unknown smoothness $\ro\in\n_{ßte 0}$ where the mean-square derivative $X^{(\ro)}$ is supposed to be H\"{o}lder continuous in quadratic mean. First, from the discrete observations $X(t_1), \dotsc, X(t_n)$, we study reconstruction of $X(t)$, $t\in[0,1]$ with $\widetilde{X}_r(t)$, a piecewise polynomial interpolation of degree $r\ge 1$. We show that the mean-square error of interpolation is a decreasing function of $r$ but becomes stable as soon as $r\ge \ro$. Next, from an interpolation-based empirical criterion, we derive an estimator $\widehat{r}$ of $\ro$ and prove its strong consistency by giving an exponential inequality for $P(\widehat{r}\not=\ro)$. Finally, we prove the strong consistency of $\widetilde{X}_{\widehat{r}}(t)$ with an almost optimal rate.