Quantile regression with generalized multiquadric loss function

TL;DR

This paper introduces a smooth, convex loss function based on multiquadric functions for quantile regression, enabling efficient gradient-based optimization and providing theoretical guarantees.

Contribution

It proposes a novel multiquadric-based loss function for quantile regression that is globally convex and suitable for gradient descent methods, improving computational efficiency.

Findings

The new loss function is globally convex and differentiable.

Gradient descent methods effectively estimate quantile regression parameters.

Simulation results show competitive performance with existing methods.

Abstract

Quantile regression (QR) is now widely used to analyze the effect of covariates on the conditional distribution of a response variable. It provides a more comprehensive picture of the relationship between a response and covariates compared with classical least squares regression. However, the non-differentiability of the check loss function precludes the use of gradient-based methods to solve the optimization problem in quantile regression estimation. To this end, This paper constructs a smoothed loss function based on multiquadric (MQ) function. The proposed loss function leads to a globally convex optimization problem that can be efficiently solved via (stochastic) gradient descent methods. As an example, we apply the Barzilai-Borwein gradient descent method to obtain the estimation of quantile regression. We establish the theoretical results of the proposed estimator under some regularity conditions, and compare it with other estimation methods using Monte Carlo simulations.