Table 2 Simulation example 2: effect of correlation between mandatory and irrelevant predictors

ρ ₀=0.25

Ridge

1.671 (0.012)

Lasso

1.911 (0.022)

0.383 (0.034)

0.100 (0.032)

0.200 (0.028)

0.975 (0.008)

Elastic net

1.744 (0.019)

0.585 (0.015)

0.400 (0.054)

0.600 (0.050)

0.835 (0.036)

\(\mathcal {M}\)-unpenalized lasso

1.741 (0.028)

0.742 (0.012)

1.000 (0.000)

0.200 (0.037)

0.938 (0.003)

\(\mathcal {M}\)-unpenalized elastic net

1.657 (0.017)

0.757 (0.008)

1.000 (0.000)

0.500 (0.064)

0.833 (0.022)

Ridle

1.492 (0.031)

0.773 (0.006)

1.000 (0.000)

0.200 (0.048)

0.931 (0.006)

ρ ₀=0.5

Ridge

1.807 (0.014)

Lasso

2.045 (0.035)

0.571 (0.013)

0.300 (0.046)

0.400 (0.039)

0.925 (0.007)

Elastic net

1.773 (0.034)

0.667 (0.008)

0.600 (0.014)

0.800 (0.048)

0.756 (0.020)

\(\mathcal {M}\)-unpenalized lasso

1.922 (0.044)

0.794 (0.003)

1.000 (0.000)

0.400 (0.047)

0.929 (0.004)

\(\mathcal {M}\)-unpenalized elastic net

1.729 (0.040)

0.796 (0.007)

1.000 (0.000)

0.700 (0.048)

0.785 (0.022)

Ridle

1.438 (0.057)

0.852 (0.006)

1.000 (0.000)

0.600 (0.049)

0.900 (0.004)

ρ ₀=0.75

Ridge

1.564 (0.022)

Lasso

1.365 (0.029)

0.684 (0.008)

0.400 (0.032)

0.600 (0.012)

0.900 (0.003)

Elastic net

1.237 (0.030)

0.745 (0.005)

0.700 (0.048)

0.900 (0.011)

0.775 (0.014)

\(\mathcal {M}\)-unpenalized lasso

1.423 (0.037)

0.839 (0.005)

1.000 (0.000)

0.700 (0.026)

0.904 (0.006)

\(\mathcal {M}\)-unpenalized elastic net

1.310 (0.041)

0.847 (0.005)

1.000 (0.000)

0.800 (0.012)

0.840 (0.008)

Ridle

0.886 (0.029)

0.875 (0.003)

1.000 (0.000)

0.700 (0.038)

0.908 (0.003)