Noncrossing structured additive multiple-output Bayesian quantile regression models

TL;DR

This paper introduces a flexible Bayesian multivariate quantile regression model with a structured additive framework, ensuring noncrossing quantiles across directions, and demonstrates its application on inequality and educational data.

Contribution

It proposes a novel Bayesian multivariate quantile regression approach with noncrossing properties using Gaussian processes, extending univariate quantile methods to multivariate responses.

Findings

Successfully models multivariate responses with noncrossing quantiles.

Applied to inequality and educational data with meaningful insights.

Demonstrates the effectiveness of the directional approach in multivariate settings.

Abstract

Quantile regression models are a powerful tool for studying different points of the conditional distribution of univariate response variables. Their multivariate counterpart extension though is not straightforward, starting with the definition of multivariate quantiles. We propose here a flexible Bayesian quantile regression model when the response variable is multivariate, where we are able to define a structured additive framework for all predictor variables. We build on previous ideas considering a directional approach to define the quantiles of a response variable with multiple-outputs and we define noncrossing quantiles in every directional quantile model. We define a Markov Chain Monte Carlo (MCMC) procedure for model estimation, where the noncrossing property is obtained considering a Gaussian process design to model the correlation between several quantile regression models. We illustrate the results of these models using two data sets: one on dimensions of inequality in the population, such as income and health; the second on scores of students in the Brazilian High School National Exam, considering three dimensions for the response variable.