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From Dense to Sparse Design: Optimal Rates under the Supremum Norm in Functional Data Analysis

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2025-09-17

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Abstract

We establish minimax-optimal rates of convergence in the supremum norm for estimating key components of functional data from repeatedly observed stochastic processes at discrete, synchronous design points - namely the mean function, the covariance kernel, and their partial derivatives. The supremum norm is employed as it controls the estimation error uniformly over the entire interval and forms the basis for constructing uniform confidence bands. The smoothness of the mean function, the stochastic processes and the covariance kernel is characterised by Hölder classes. Under dense designs, meaning sufficiently many design points per curve, the estimators for the mean function and the covariance kernel achieve the parametric √n rate without additional logarithmic factors and corresponding central limit theorems are derived. In sparse settings, the discretisation error dominates while between the sparse and dense regimes an additional regime appears, driven by measurement errors which is unique to the use of the supremum norm. For the estimation of partial derivatives sufficiently smooth sample paths of the stochastic processes are crucial to obtain the √n rate, whereas paths of lower order differentiability necessarily lead to a slower rate of convergence in the dense regime. The theoretical analysis is conducted with abstract weights; however, we show that local polynomial estimators satisfy the required assumptions and their practical performance is demonstrated through simulations and real-data applications.

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Dates
Issued: 2025-09-17
Faculty
FB12:Mathematik und Informatik
Language
en
Keywords
FDAfunctional data analysismean functioncovariance kernelpartial derivativesdense and sparse designminimax optimalitysupremum normasymptotic normalityoptimal rates of convergencesynchronously sampled datasmoothness of the sample paths
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Attribution-NonCommercial-NoDerivatives 4.0 International
URI
https://open.uni-marburg.de/handle/openumr/9232
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Berger, Max Jannis: From Dense to Sparse Design: Optimal Rates under the Supremum Norm in Functional Data Analysis. : 2025-09-17.

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Attribution-NonCommercial-NoDerivatives 4.0 International
Except where otherwised noted, this item's license is described as Attribution-NonCommercial-NoDerivatives 4.0 International