A Critical Review of LASSO and Its Derivatives for Variable Selection Under Dependence Among Covariates

The limitations of the well-known LASSO regression as a variable selector are tested when there exists dependence structures among covariates. We analyse both the classic situation with n >= p and the high dimensional framework with p > n. Known restrictive properties of this methodology to guarantee optimality, as well as inconveniences in practice, are analysed and tested by means of an extensive simulation study. Examples of these drawbacks are showed 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 work best in terms of the considered data nature.