  Whole robustness is a nice property to have for statistical models. It
implies that the impact of outliers gradually vanishes as they approach plus or
minus infinity. So far, the Bayesian literature provides results that ensure
whole robustness for the location-scale model. In this paper, we make two
contributions. First, we generalise the results to attain whole robustness in
simple linear regression through the origin, which is a necessary step towards
results for general linear regression models. We allow the variance of the
error term to depend on the explanatory variable. This flexibility leads to the
second contribution: we provide a simple Bayesian approach to robustly estimate
finite population means and ratios. The strategy to attain whole robustness is
simple since it lies in replacing the traditional normal assumption on the
error term by a super heavy-tailed distribution assumption. As a result, users
can estimate the parameters as usual, using the posterior distribution.
