Abstract: The enveloping approach employs sufficient dimension-reduction techniques to gain estimation efficiency, and has been used in several multivariate analysis contexts. However, its Bayesian development has been sparse, and the only Bayesian envelope construction is in the context of a linear regression. In this paper, we propose a Bayesian envelope approach to a quantile regression, using a general framework that may potentially aid enveloping in other contexts as well. The proposed approach is also extended to accommodate censored data. Data augmentation Markov chain Monte Carlo algorithms are derived for approximate sampling from the posterior distributions. Simulations and data examples are included for illustration.

Key words and phrases: Envelope model, metropolis-within-gibbs sampling, quantile regression, sufficient dimension reduction, tobit quantile.