Adaptive estimation of irregular mean and covariance functions

TL;DR

This paper introduces adaptive nonparametric estimators for mean and covariance functions of functional data, accommodating irregularity, measurement error, and various sampling designs, with demonstrated effectiveness through simulations.

Contribution

It proposes a novel adaptive estimation method that adjusts to local regularity and handles heteroscedastic errors in functional data analysis.

Findings

Estimators perform well in simulations with irregular, noisy data.

Method adapts to unknown local regularity of the process.

Effective for both sparse and dense sampling scenarios.

Abstract

Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.