Rank theory approach to ridge, LASSO, preliminary test and Stein‐type estimators: A comparative study

Abstract

AbstractIn 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 L2‐risk of the restricted R‐estimator equals the lower bound on the L2‐risk of LASSO. Our conclusions are based on L2‐risk analysis and relative L2‐risk efficiencies with related tables and graphs. The Canadian Journal of Statistics 46: 690–704; 2018 © 2018 Société statistique du Canada