Geometric ergodicity of the Gibbs sampler for Bayesian quantile regression

Consider the quantile regression model Y = X β + σ ϵ where the components of ϵ are i.i.d. errors from the asymmetric Laplace distribution with r th quantile equal to 0, where r ∈ ( 0 , 1 ) is fixed. Kozumi and Kobayashi (2011) [9] introduced a Gibbs sampler that can be used to explore the intractable posterior density that results when the quantile regression likelihood is combined with the usual normal/inverse gamma prior for ( β , σ ) . In this paper, the Markov chain underlying Kozumi and Kobayashi’s (2011) [9] algorithm is shown to converge at a geometric rate. No assumptions are made about the dimension of X , so the result still holds in the “large p , small n ” case.