High dimensional test for functional covariates

TL;DR

This paper introduces a new statistical testing method for analyzing the relationship between outcomes and high-dimensional functional predictors, with applications in medical data analysis.

Contribution

It develops a novel inference procedure for high-dimensional functional linear models, including estimation, hypothesis testing, and theoretical guarantees.

Findings

The proposed method achieves accurate estimation in high-dimensional settings.

It successfully tests hypotheses on functional parameters with theoretical guarantees.

Application to Alzheimer's data demonstrates practical utility.

Abstract

As medical devices become more complex, they routinely collect extensive and complicated data. While classical regressions typically examine the relationship between an outcome and a vector of predictors, it becomes imperative to identify the relationship with predictors possessing functional structures. In this article, we introduce a novel inference procedure for examining the relationship between outcomes and large-scale functional predictors. We target testing the linear hypothesis on the functional parameters under the generalized functional linear regression framework, where the number of the functional parameters grows with the sample size. We develop the estimation procedure for the high dimensional generalized functional linear model incorporating B-spline functional approximation and amenable regularization. Furthermore, we construct a procedure that is able to test the local alternative hypothesis on the linear combinations of the functional parameters. We establish the statistical guarantees in terms of non-asymptotic convergence of the parameter estimation and the oracle property and asymptotic normality of the estimators. Moreover, we derive the asymptotic distribution of the test statistic. We carry out intensive simulations and illustrate with a new dataset from an Alzheimer's disease magnetoencephalography study.