  Linear regression with the classical normality assumption for the error
distribution may lead to an undesirable posterior inference of regression
coefficients due to the potential outliers. This paper considers the finite
mixture of two components with thin and heavy tails as the error distribution,
which has been routinely employed in applied statistics. For the heavily-tailed
component, we introduce the novel class of distributions; their densities are
log-regularly varying and have heavier tails than those of Cauchy distribution,
yet they are expressed as a scale mixture of normal distributions and enable
the efficient posterior inference by Gibbs sampler. We prove the robustness to
outliers of the posterior distributions under the proposed models with a
minimal set of assumptions, which justifies the use of shrinkage priors with
unbounded densities for the coefficient vector in the presence of outliers. The
extensive comparison with the existing methods via simulation study shows the
improved performance of our model in point and interval estimation, as well as
its computational efficiency. Further, we confirm the posterior robustness of
our method in the empirical study with the shrinkage priors for regression
coefficients.
