Abstract

The covariance function is a central object in functional and longi

tudinal data analysis, yet its optimal estimation remains challenging because

of its bivariate structure and its sensitivity to the sampling design. We study

covariance function estimation for functional data under two distinct designs:

the common design, in which all curves are observed on a shared grid, and the

independent design, in which the sampling locations vary across curves. By exploiting the tensor-product structure of the covariance function, we propose a

unified smoothing-spline estimator that applies to both settings.

We establish minimax optimal convergence rates for the proposed estimator

under both sampling designs and identify several phase transition phenomena.

In particular, under the independent design with sparse sampling, the estimator

attains a nearly “univariate” minimax rate of (nm)−2α/(2α+1), up to a logarithmic factor, despite the intrinsically bivariate nature of the covariance function.

By contrast, under the common design, the sparse-regime rate is dominated by

∗The authors are listed in alphabetical order by surname.

the discretization error m−2α. We further develop an adaptive implementation

based on cross-validation for selecting the smoothing parameter, which does not

require prior knowledge of the smoothness parameter α or the sampling-frequency

regime. Numerical experiments on both simulated and real datasets support the

theoretical findings and demonstrate the practical effectiveness of the proposed

method.

Key words and phrases: Covariance function, functional data analysis, method of regularization, minimax rate, phase transition, Sobolev space, tensor product space

Information

Preprint No.SS-2026-0082
Manuscript IDSS-2026-0082
Complete AuthorsT. Tony Cai, Yicheng Li, Ming Yuan
Corresponding AuthorsT. Tony Cai
Emailstcai@wharton.upenn.edu

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Acknowledgments

The authors thank the editor and the referees for their helpful comments

and suggestions.

Supplementary Materials

Detailed proofs, computational details, and additional numerical results are

provided in the supplementary material.

Supplementary materials are available for download.