Bayesian joint-quantile regression

Estimation of low or high conditional quantiles is called for in many applications, but commonly encountered data sparsity at the tails of distributions makes this a challenging task. We develop a Bayesian joint-quantile regression method to borrow information across tail quantiles through a linear approximation of quantile coefficients. Motivated by a working likelihood linked to the asymmetric Laplace distributions, we propose a new Bayesian estimator for high quantiles by using a delayed rejection and adaptive Metropolis and Gibbs algorithm. We demonstrate through numerical studies that the proposed estimator is generally more stable and efficient than conventional methods for estimating tail quantiles, especially at small and modest sample sizes.