Mean and Covariance Estimation for Functional Snippets

TL;DR

This paper introduces a hybrid method for estimating mean and covariance functions of functional snippets, addressing challenges posed by incomplete data and missing information in off-diagonal regions.

Contribution

It proposes a novel decomposition approach combining nonparametric variance estimation with parametric correlation modeling for functional snippets.

Findings

Effective covariance estimation demonstrated through simulations

New estimator for measurement error variance with proven asymptotic properties

Hybrid strategy improves estimation accuracy in incomplete data scenarios

Abstract

We consider estimation of mean and covariance functions of functional snippets, which are short segments of functions possibly observed irregularly on an individual specific subinterval that is much shorter than the entire study interval. Estimation of the covariance function for functional snippets is challenging since information for the far off-diagonal regions of the covariance structure is completely missing. We address this difficulty by decomposing the covariance function into a variance function component and a correlation function component. The variance function can be effectively estimated nonparametrically, while the correlation part is modeled parametrically, possibly with an increasing number of parameters, to handle the missing information in the far off-diagonal regions. Both theoretical analysis and numerical simulations suggest that this hybrid strategy % divide-and-conquer strategy is effective. In addition, we propose a new estimator for the variance of measurement errors and analyze its asymptotic properties. This estimator is required for the estimation of the variance function from noisy measurements.