Global smoothness estimation of a Gaussian process from regular sequence designs

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.Comment: 28 page

Assessing the number of mean-square derivatives of a Gaussian process

28 pagesInternational audienceWe 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

Local superefficiency of data-driven projection density estimators in continuous time

We construct a data-driven projection density estimator for continuous time processes. This estimator reaches superoptimal rates over a class F0 of densities that is dense in the family of all possible densities, and a «reasonable» rate elsewhere. The class F0 may be chosen previously by the analyst. Results apply to Rd- Rd-valued processes and to N-valued processes. In the particular case where squareintegrable local time does exist, it is shown that our estimator is strictly better than the local time estimator over F0

Modelization and Nonparametric estimation for a dynamical system with noise

International audienceWe examine the effect of two specific noises on a dynamical system. We obtain consistent estimates with their rates of convergence for the invariant density for such a model. Some simulations are provided