Consistent and unbiased variable selection under indepedent features using Random Forest permutation importance

Variable selection in sparse regression models is an important task as applications ranging from biomedical re-search to econometrics have shown. Especially for higher dimensional regression problems, for which the regres-sion function as the link between response and covariates cannot be directly detected, the selection of informative variables is challenging. Under these circumstances, the Random Forest method is an helpful tool to predict new outcomes while delivering measures for variable selection. One common approach is the usage of the permutation importance. Due to its intuitive idea and flexible usage, it is important to explore circumstances, for which the permutation importance based on Random Forest correctly indicates informative covariates. Regarding the latter, we deliver theoretical guarantees for the validity of the permutation importance measure under specific assump-tions such as the mutual independence of the features and prove its (asymptotic) unbiasedness, while under slightly stricter assumptions, consistency of the permutation importance measure is established. An extensive simulation study supports our findings.