High‐Dimensional Multiresponse Partially Functional Linear Regression

Abstract

ABSTRACT We propose a new class of high‐dimensional multiresponse partially functional linear regressions (MR‐PFLRs) to investigate the relationship between scalar responses and a set of explanatory variables, which include both functional and scalar types. In this framework, both the dimensionality of the responses and the number of scalar covariates can diverge to infinity. To account for within‐subject correlation, we develop a functional principal component analysis (FPCA)‐based penalized weighted least squares estimation procedure. In this approach, the precision matrix is estimated using penalized likelihoods, and the regression coefficients are then estimated through the penalized weighted least squares method, with the precision matrix serving as the weight. This method allows for the simultaneous estimation of both functional and scalar regression coefficients, as well as the precision matrix, while identifying significant features. Under mild conditions, we establish the consistency, rates of convergence, and oracle properties of the proposed estimators. Simulation studies demonstrate the finite‐sample performance of our estimation method. Additionally, the practical utility of the MR‐PFLR model is showcased through an application to Alzheimer's disease neuroimaging initiative (ADNI) data.