Abstract

We propose a functional linear operator quantile regression (FLOQR) framework, which in

cludes many important and useful functional data models, and devote to the new framework model for

longitudinal data with the typically sparse and irregular designs. The non-smooth quantile loss and

functional linear operator pose new challenges to functional data analysis for longitudinal data in both

computation and theoretical development. To address the challenge, we propose the iterative surrogate

least squares estimation approach for the FLOQR model, which transforms the response trajectories

and establishes a new connection between FLOQR and functional linear operator model. In addition,

we use Karhunen-Loève expansion to alleviate the problem of the nonexistence of the inverse of the

covariance in the infinite-dimensional Hilbert space. Then, the approach is used to classic functional

varying coefficient QR, functional linear QR, and functional varying coefficient QR with history index

function for sparse longitudinal data by using functional principal components analysis through conditional expectation. The resulting technique is flexible and allows the prediction of an unobserved

quantile response trajectory from sparse measurements of a predictor trajectory.

Theoretically, we

show that, after a constant number of iterations, the proposed estimator is asymptotic consistent for

sparse designs. Moreover, asymptotic pointwise confidence bands are obtained for predicted quantile

individual trajectories based on their asymptotic distributions. The proposed algorithms perform well

in simulations, and are illustrated with longitudinal primary biliary liver cirrhosis data.

Information

Preprint No.SS-2023-0415
Manuscript IDSS-2023-0415
Complete AuthorsXingcai Zhou, Tingyu Lai, Linglong Kong
Corresponding AuthorsLinglong Kong
Emailslkong@ualberta.ca

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Acknowledgments

Dr. Zhou was supported by the National Natural Science Foundation of China (12171242,

12371267). Dr. Lai was supported by the National Natural Science Foundation of China

under Grant 12271014. Dr. Kong was partially supported by grants from the Canada CIFAR

AI Chairs program, the Alberta Machine Intelligence Institute (AMII), the Natural Sciences

and Engineering Council of Canada (NSERC), and the Canada Research Chair program from

NSERC.

Supplementary Materials

The online Supplementary Material includes all proofs, technical details and additional experimental results.

Supplementary materials are available for download.