A. K. Md. Ehsanes Saleh - Rank-Based Methods for Shrinkage and Selection

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1.1 Four plots using different versions of the telephone data set with fitted lines.

1.2 Histograms and ordered residual plots of LS and Theil estimators.

1.3 Effect of a single outlier on LS and rank estimators.

1.4 Gradients of absolute value (Bn′(θ)) and dispersion (Dn′(θ)) functions.

1.5 Scoring functions ϕ(u)=12(u−0.5) and ϕ+(u)=3u.

1.6 Dispersion functions and derivative plots for 1.1(d).

1.7 Key shrinkage characteristics of LASSO and ridge.

1.8 Geometric interpretation of ridge.

1.9 Geometric interpretation of LASSO.

2.1 The first-order nature of shrinkage due to ridge.

2.2 Two outliers found in the Q–Q plot for the Swiss data set.

2.3 Sampling distributions of rank estimates.

2.4 Shrinkage of β 5due to increase in ridge tuning parameter, λ 2.

2.5 Ridge traces for orthonormal, diagonal, LS, and rank estimators ( m = 40).

2.6 MSE Derivative plot to find optimal λ 2for the diagonal case.

2.7 Bias, variance and MSE for the Swiss data set (optimal λ 2= 70.8).

2.8 MSE for training, CV and test sets, and coefficients from the ridge trace.

2.9 The first-order nature of shrinkage due to LASSO.

2.10 Diamond-warping effect of weights in the aLASSO estimator for p = 2.

2.11 Comparison of LASSO and aLASSO traces for the Swiss data set.

2.12 Variable ordering from R-LASSO and R-aLASSO traces for the Swiss data set.

2.13 Ranked residuals of the diabetes data set. (Source: Rfit() package in R.)

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