# Basis Expansions for Functional Snippets

## TL;DR

This paper develops a basis expansion method for estimating mean and covariance functions from functional snippets, where data are only available on short intervals, enabling consistent estimation despite limited data coverage.

## Contribution

It introduces a novel basis representation approach for covariance estimation from snippets, addressing the challenge of limited data coverage and establishing convergence rates.

## Key findings

- Consistent covariance estimation from short-interval snippets.
- The proposed method achieves favorable finite-sample performance.
- Simulation and real data demonstrate effectiveness.

## Abstract

Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. In this paper, we investigate mean and covariance estimation for functional snippets in which observations from a subject are available only in an interval of length strictly (and often much) shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via a basis representation of the covariance function. The proposed approach allows one to consistently estimate an infinite-rank covariance function from functional snippets. We establish the convergence rates for the proposed estimators and illustrate their finite-sample performance via simulation studies and two data applications.