A critical review of LASSO and its derivatives for variable selection under dependence among covariates

We study the limitations of the well known LASSO regression as a variable selector when there exists dependence structures among covariates. We analyze both the classic situation with $n\ge p$ and the high dimensional framework with $p>n$. Restrictive properties of this methodology to guarantee optimality, as well as the inconveniences in practice, are analyzed. Examples of these drawbacks are showed by means of a extensive simulation study, making use of different dependence scenarios. In order to search for improvements, a broad comparison with LASSO derivatives and alternatives is carried out. Eventually, we give some guidance about what procedures are the best in terms of the data nature.