arXiv subjects

Max Berger

Publications and source records attributed to Max Berger.

3 recordsLinked to original sources

arXiv2025

Smooth and rough paths in mean derivative estimation for functional data

In this paper, in a multivariate setting we derive near optimal rates of convergence in the minimax sense for estimating partial derivatives of the mean function for functional data observed under a fixed synchronous design over H\"older smoothness classes. We focus on the supremum norm since it corresponds to the visualisation of the estimation error, and is closely related to the construction of uniform confidence bands. In contrast to mean function estimation, for derivative estimation the smoothness of the paths of the processes is crucial for the rates of convergence. On the one hand, if the paths have higher-order smoothness than the order of the partial derivative to be estimated, the parametric $\sqrt n$ rate can be achieved under sufficiently dense design. On the other hand, for processes with rough paths of lower-order smoothness, we show that the rates of convergence are necessarily slower than the parametric rate, and determine a near-optimal rate at which estimation is still possible. We implement a multivariate local polynomial derivative estimator and illustrate its finite-sample performance in a simulation as well as for two real-data sets. To assess the smoothness of the sample paths in the applications we further discuss a method based on comparing restricted estimates of the partial derivatives of the covariance kernel.

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arXiv2024

Optimal rates for estimating the covariance kernel from synchronously sampled functional data

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 $L_2$ 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 $\sqrt n$-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.

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arXiv2023

From dense to sparse design: Optimal rates under the supremum norm for estimating the mean function in functional data analysis

We derive optimal rates of convergence in the supremum norm for estimating the H\"older-smooth mean function of a stochastic process which is repeatedly and discretely observed with additional errors at fixed, multivariate, synchronous design points, the typical scenario for machine recorded functional data. Similarly to the optimal rates in $L_2$ obtained in \citet{cai2011optimal}, for sparse design a discretization term dominates, while in the dense case the parametric $\sqrt n$ rate can be achieved as if the $n$ processes were continuously observed without errors. The supremum norm is of practical interest since it corresponds to the visualization of the estimation error, and forms the basis for the construction uniform confidence bands. We show that in contrast to the analysis in $L_2$, there is an intermediate regime between the sparse and dense cases dominated by the contribution of the observation errors. Furthermore, under the supremum norm interpolation estimators which suffice in $L_2$ turn out to be sub-optimal in the dense setting, which helps to explain their poor empirical performance. In contrast to previous contributions involving the supremum norm, we discuss optimality even in the multivariate setting, and for dense design obtain the $\sqrt n$ rate of convergence without additional logarithmic factors. We also obtain a central limit theorem in the supremum norm, and provide simulations and real data applications to illustrate our results.

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