# Robust regression for heavy-tailed random design

As far as I know, there are robust regression methods for outliers in response $$Y$$ and heavy-tailed error $$\epsilon$$. The settings for the design matrix (predictor) $$X$$ is either fixed design or sub-exponential like Normal distribution. What if the design matrix $$X$$ is random design and the distribution is heavy-tailed? Is there any advanced methods apart from truncation on $$X$$ or $$\log X$$?

One implementation is lqs in R package MASS, see the companion book by Venables & Ripley. For code example see How to get summary statistics from "resistant regression" - lqs - in R?
One simple method would be to use the quantiles of $$X$$ rather than $$X$$ per se as predictor, e.g. adding a column which equals $$1$$ if $$x_k$$ is larger than the 90 quantile of $$X$$. This avoids truncation or arbitrarily changing the distribution of $$X$$.