Quantile Regression Modelling via Location and Scale Mixtures of Normal Distributions

TL;DR

This paper introduces an EM algorithm-based approach for quantile regression using location and scale mixtures of normal distributions, enabling efficient estimation, variance calculation, and extension to mixed effects models.

Contribution

It presents a novel EM algorithm for quantile regression with stable variance estimation and extends to mixed effects models, improving computational efficiency and diagnostic capabilities.

Findings

The EM algorithm effectively estimates quantile regression coefficients.

Variance-covariance matrices are more stable with the proposed kernel density estimator.

The methodology outperforms existing software in simulations and case studies.

Abstract

We show that the estimating equations for quantile regression can be solved using a simple EM algorithm in which the M-step is computed via weighted least squares, with weights computed at the E-step as the expectation of independent generalized inverse-Gaussian variables. We compute the variance-covariance matrix for the quantile regression coefficients using a kernel density estimator that results in more stable standard errors than those produced by existing software. A natural modification of the EM algorithm that involves fitting a linear mixed model at the M-step extends the methodology to mixed effects quantile regression models. In this case, the fitting method can be justified as a generalized alternating minimization algorithm. Obtaining quantile regression estimates via the weighted least squares method enables model diagnostic techniques similar to the ones used in the linear regression setting. The computational approach is compared with existing software using simulated data, and the methodology is illustrated with several case studies.