Rank theory approach to ridge, LASSO, preliminary test and Stein-type estimators: a comparative study
Abstract
In the development of efficient predictive models, the key is to identify suitable predictors to establish a prediction model for a given linear or nonlinear model. This paper provides a comparative study of ridge regression, least absolute shrinkage and selector operator (LASSO), preliminary test (PTE) and Stein-type estimators based on the theory of rank statistics. Under the orthonormal design matrix of a given linear model, we find that the rank-based ridge estimator outperforms the usual rank estimator, restricted R-estimator, rank-based LASSO, PTE and Stein-type R-estimators uniformly. On the other hand, neither LASSO nor the usual R-estimator, preliminary test and Stein-type R-estimators outperform the other. The region of dominance of LASSO over all the R-estimators (except the ridge R-estimator) is the sparsity-dimensional interval around the origin of the parameter space. We observe that the $L_2$-risk of the restricted R-estimator equals the lower bound on the $L_2$-risk of LASSO. Our conclusions are based on $L_2$-risk analysis and relative $L_2$-risk efficiencies with related tables and graphs.
