An Alternative Approach to Functional Linear Partial Quantile Regression

We have previously proposed the partial quantile regression (PQR) prediction procedure for functional linear model by using partial quantile covariance techniques and developed the simple partial quantile regression (SIMPQR) algorithm to efficiently extract PQR basis for estimating functional coefficients. However, although the PQR approach is considered as an attractive alternative to projections onto the principal component basis, there are certain limitations to uncovering the corresponding asymptotic properties mainly because of its iterative nature and the non-differentiability of the quantile loss function. In this article, we propose and implement an alternative formulation of partial quantile regression (APQR) for functional linear model by using block relaxation method and finite smoothing techniques. The proposed reformulation leads to insightful results and motivates new theory, demonstrating consistency and establishing convergence rates by applying advanced techniques from empirical process theory. Two simulations and two real data from ADHD-200 sample and ADNI are investigated to show the superiority of our proposed methods.