Combining information both within and across trajectories, we propose a simple estimator for the local regularity of the trajectories of a stochastic process. Independent trajectories are measured with errors at randomly sampled time points. The proposed approach is model-free and applies to a large class of stochastic processes. Non-asymptotic bounds for the concentration of the estimator are derived. Given the estimate of the local regularity, we build a nearly optimal local polynomial smoother from the curves from a new, possibly very large sample of noisy trajectories. We derive non-asymptotic pointwise risk bounds uniformly over the new set of curves. Our estimates perform well in simulations, in both cases of differentiable or non-differentiable trajectories. Real data sets illustrate the effectiveness of the new approaches.
The authors thank Groupe Renault and the ANRT (French National Association for Research and Technology) for their financial support via the CIFRE convention no. 2017/1116. Valentin Patilea gratefully acknowledges support from the Joint Research Initiative “Models and mathematical processing of very large data” under the aegis of Risk Foundation, in partnership with MEDIAMETRIE and GENES, France.
We thank the Associate Editor and an anonymous reviewer for their careful reading and constructive comments, which helped us to improve the manuscript.
"Learning the smoothness of noisy curves with application to online curve estimation." Electron. J. Statist. 16 (1) 1485 - 1560, 2022. https://doi.org/10.1214/22-EJS1997