Simultaneous Bayesian Estimation of Multiple Quantiles with an Extension to Hierarchical Models

Abstract

Simultaneously modeling multiple quantiles by possibly incorporating constraints across quantiles, in particular that of monotonicity, has been an important problem. While recent attempts to address this problem focus mostly on the monotonicity issue, we take a different route using a Bayesian approach. We propose a parametric pseudo-likelihood based approach for simultaneous Bayesian estimation of multiple quantiles that is computationally simple and has the flexibility to accomodate linear as well as nonlinear model forms along with different types of prior specifications. A unique feature of our method compared to existing approaches is the posterior consistency property for the case of linear quantile regression. Further, we develop a useful extension of our method to a hierarchical setting which is applicable in particular to the normal random effects model and binary regression. We demonstrate our methods using simulations and two real life examples. The first example demonstrates an alternative way to address heteroskedasticity issues in modeling worker’s compensation claims. The second example provides a novel approach to flexibly model inefficiencies of firms in a stochastic frontier analysis applied to a dataset on hospital costs.