Learning the smoothness of noisy curves with application to online curve estimation

Combining information both within and across trajectories, we propose a simple estimator for the local regularity of trajectories stochastic process. Independent are measured with errors at randomly sampled time points. The proposed approach is model-free applies to large class processes. Non-asymptotic bounds concentration derived. Given estimate regularity, build nearly optimal polynomial smoother from curves new, possibly very sample noisy trajectories. We derive non-asymptotic pointwise risk uniformly over new set curves. Our estimates perform well in simulations, cases differentiable or non-differentiable Real data sets illustrate effectiveness approaches.