Table II.

Ridge penalty: nine covariates, variations of ρ (N =5000).

Penalized estimates*	ρ †	λ ‡	Bias§	Empirical precision¶	Accuracy of precision∥	MSE**
	0	0.0	−0.023	0.058	0.006	0.058
	0.9	0.0	−0.022	0.138	0.024	0.139
${\widehat{\beta }}_{1j}$	0	0.1	−0.046	0.054	0.008	0.056
	0.9	0.2	−0.128	0.109	0.031	0.125
${\widehat{\beta }}_{\mathit{pj}}$, p=(1,…,9)	0	0.4	−0.071	0.048	−0.002	0.054
	0.9	0.4	−0.101	0.127	0.004	0.137

^*Values for bias, precision, etc. refer to ${\widehat{\beta }}_{11}$, where the simulated value of β₁₁ is 1.79.

^†Bivariate correlations among covariates.

^‡Candidate value for tuning parameter. Results are shown for λ=0 and the candidate λ value which minimized R, the cross-validated log-likelihood loss for the largest number of simulated data sets.

^§${\beta }_{11}-\left[\right(1∕\mathit{nsi}\left){\Sigma }_{\mathit{si}=1}^{\mathit{nsi}}{\widehat{\beta }}_{11,\mathit{si},\lambda }\right]$, where si are simulation iterations.

^¶${\Sigma }_{\mathit{si}=1}^{\mathit{nsi}}{({\widehat{\beta }}_{11,\mathit{si},\lambda }-{\beta }_{11,\lambda })}^2∕(n-1)$.

^∥$(1∕\mathit{nsi}){\Sigma }_{\mathit{si}=1}^{\mathit{nsi}}v\widehat{a}r\left({\widehat{\beta }}_{11,\mathit{si},\lambda }\right)$—empirical precision_λ.

^**Mean square error = empirical ${\text{precision}}_{\lambda }+{\text{bias}}_{\lambda }^2$.