Joint estimation of quantile planes over arbitrary predictor spaces

TL;DR

This paper introduces a new Bayesian approach for joint estimation of non-crossing quantile planes over arbitrary convex predictor spaces, enabling efficient computation and improved accuracy in quantile regression.

Contribution

It proposes a novel parametrization for non-crossing quantile planes and develops a Bayesian methodology with Gaussian process priors for flexible, efficient estimation.

Findings

Method shows better accuracy than existing approaches

Offers improved coverage and model fit

Ensures posterior consistency under mild conditions

Abstract

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parametrization that characterizes any collection of non-crossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parametrization inherit a fast computation of the likelihood function, enabling penalized likelihood or Bayesian approaches to model fitting. We introduce a complete Bayesian methodology by using Gaussian process prior distributions on the function valued parameters and develop a robust and efficient Markov chain Monte Carlo parameter estimation. The resulting method is shown to offer posterior consistency under mild tail and regularity conditions. We present several illustrative examples where the new method is compared against existing approaches and is found to offer better accuracy, coverage and model fit.