Inference for Sparse and Dense Functional Data with Covariate Adjustments

TL;DR

This paper develops inference methods for covariate-adjusted functional data, distinguishing between sparse and dense data scenarios, and introduces finite-sample corrections to improve inference accuracy.

Contribution

It provides a double asymptotic framework for functional data with covariate adjustments and proposes finite-sample corrections for more reliable inference.

Findings

Existing asymptotic normality results can be misleading in finite samples

Finite-sample corrections improve inference accuracy

Theoretical results are validated with real-data application

Abstract

We consider inference for the mean and covariance functions of covariate adjusted functional data using Local Linear Kernel (LLK) estimators. By means of a double asymptotic, we differentiate between sparse and dense covariate adjusted functional data - depending on the relative order of m (the discretization points per function) and n (the number of functions). Our simulation results demonstrate that the existing asymptotic normality results can lead to severely misleading inferences in finite samples. We explain this phenomenon based on our theoretical results and propose finite-sample corrections which provide practically useful approximations for inference in sparse and dense data scenarios. The relevance of our theoretical results is showcased using a real-data application.