  In this article, we propose new Bayesian methods for selecting and estimating
a sparse coefficient vector for skewed heteroscedastic response. Our novel
Bayesian procedures effectively estimate the median and other quantile
functions, accommodate non-local prior for regression effects without
compromising ease of implementation via sampling based tools, and
asymptotically select the true set of predictors even when the number of
covariates increases in the same order of the sample size. We also extend our
method to deal with some observations with very large errors. Via simulation
studies and a re-analysis of a medical cost study with large number of
potential predictors, we illustrate the ease of implementation and other
practical advantages of our approach compared to existing methods for such
studies.
