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efficiency of LASSO; penalty estimators; preliminary test; Stein-type estimator; ridge estimator; L$_2$-risk function
In the development of efficient predictive models, the key is to identify suitable predictors for a given linear model. For the first time, this paper provides a comparative study of ridge regression, LASSO, preliminary test 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, preliminary test 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 domination of LASSO over all the R-estimators (except the ridge R-estimator) is the interval around the origin of the parameter space. Finally, 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.
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