[math] \require{textmacros} \def \bbeta {\bf \beta} \def\fat#1{\mbox{\boldmath$#1$}} \def\reminder#1{\marginpar{\rule[0pt]{1mm}{11pt}}\textbf{#1}} \def\SSigma{\bf \Sigma} \def\ttheta{\bf \theta} \def\aalpha{\bf \alpha} \def\ddelta{\bf \delta} \def\eeta{\bf \eta} \def\llambda{\bf \lambda} \def\ggamma{\bf \gamma} \def\nnu{\bf \nu} \def\vvarepsilon{\bf \varepsilon} \def\mmu{\bf \mu} \def\nnu{\bf \nu} \def\ttau{\bf \tau} \def\SSigma{\bf \Sigma} \def\TTheta{\bf \Theta} \def\XXi{\bf \Xi} \def\PPi{\bf \Pi} \def\GGamma{\bf \Gamma} \def\DDelta{\bf \Delta} \def\ssigma{\bf \sigma} \def\UUpsilon{\bf \Upsilon} \def\PPsi{\bf \Psi} \def\PPhi{\bf \Phi} \def\LLambda{\bf \Lambda} \def\OOmega{\bf \Omega} [/math]

Derive the equivalent of Equations (equation) and (equation) for the targeted ridge logistic regression estimator:

[[math]] \begin{eqnarray*} \hat{\bbeta}(\lambda, \bbeta_0) & = & \arg \min_{\bbeta \in \mathbb{R}^p} \mathbf{Y}^{\top} \mathbf{X} \hat{\bbeta}(\lambda; \bbeta_0) - \sum\nolimits_{i=1}^n \log \{ 1 + \exp[ \mathbf{X}_{i,\ast} \hat{\bbeta}(\lambda; \bbeta_0) ] \} - \tfrac{1}{2} \lambda \| \hat{\bbeta}(\lambda; \bbeta_0) - \bbeta_0 \|_2^2, \end{eqnarray*} [[/math]]

with nonrandom [math]\bbeta_0 \in \mathbb{R}^p[/math].