Inference in Functional Linear Quantile Regression

TL;DR

This paper develops a statistical inference method for functional linear quantile regression, allowing testing of whether the effect of a functional covariate on response quantiles remains constant across different quantile levels, with practical implementation and validation.

Contribution

It introduces an adjusted Wald test for hypothesis testing in functional quantile regression, combining functional principal component analysis with quantile regression, and provides an easy-to-implement R package.

Findings

The test maintains good size and power in simulations.

The method is effective with sparse, noisy data.

Application to bike share data demonstrates practical utility.

Abstract

In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest. The parameter function is estimated by combining ideas from functional principal component analysis and quantile regression. An adjusted Wald testing procedure is proposed for this hypothesis of interest, and its chi-square asymptotic null distribution is derived. The testing procedure is investigated numerically in simulations involving sparse and noisy functional covariates and in a capital bike share data application. The proposed approach is easy to implement and the {\tt R} code is published online at \url{https://github.com/xylimeng/fQR-testing}.