Geometric ergodicity of the Gibbs sampler for Bayesian quantile regression

Consider the quantile regression model Y=Xβ+σϵY=Xβ+σϵ where the components of ϵϵ are i.i.d. errors from the asymmetric Laplace distribution with rrth quantile equal to 0, where r∈(0,1)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)(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 XX, so the result still holds in the “large pp, small nn” case.