Papers › Optimal rates for estimating the covariance kernel from synchronously sampled functional data
Optimal rates for estimating the covariance kernel from synchronously sampled functional data
Max Berger, Hajo Holzmann
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We obtain minimax-optimal convergence rates in the supremum norm, including information-theoretic lower bounds, for estimating the covariance kernel of a stochastic process which is repeatedly observed at discrete, synchronous design points. We focus on the supremum norm instead of the simpler L₂ norm, since it corresponds to the visualization of the estimation error and forms the basis for the construction of uniform confidence bands. For dense design, assuming H\"older-smooth sample paths we obtain the √(n)-rate of convergence in the supremum norm without additional logarithmic factors which typically occur in the results in the literature. Surprisingly, in the transition from dense to sparse design the rates do not reflect the two-dimensional nature of the covariance kernel but correspond to those for univariate mean function estimation. Our estimation method can make use of higher-order smoothness of the covariance kernel away from the diagonal, and does not require the same smoothness on the diagonal itself. Hence, our results cover covariance kernels of processes with rough, non-differentiable sample paths. Moreover, the estimator does not use mean function estimation to form residuals, and no smoothness assumptions on the mean have to be imposed. In the dense case we also obtain a central limit theorem in the supremum norm, which can be used as the basis for the construction of uniform confidence sets. Extensions to estimating partial derivatives as well as to asynchronous designs are also discussed. Simulations and real-data applications illustrate the practical usefulness of the methods.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections