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Learning the smoothness of noisy curves with application to online curve estimation. (English) Zbl 1493.62641

Summary: 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.

MSC:

62R10 Functional data analysis
62G05 Nonparametric estimation
62M09 Non-Markovian processes: estimation

Software:

mclust; np; fda (R)
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Full Text: DOI arXiv Link

References:

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