[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]

Consider an experiment involving [math]n[/math] cancer samples. For each sample [math]i[/math] the transcriptome of its tumor has been profiled and is denoted [math]\mathbf{X}_{i} = (X_{i1}, \ldots, X_{ip})^{\top}[/math] where [math]X_{ij}[/math] represents the gene [math]j=1, \ldots, p[/math] in sample [math]i[/math]. Additionally, the overall survival data, [math](Y_i, c_i)[/math] for [math]i=1, \ldots,n[/math] of these samples is available. In this [math]Y_i[/math] denotes the survival time of sample [math]i[/math] and [math]c_i[/math] the event indicator with [math]c_i = 0[/math] and [math]c_i = 1[/math] representing non- and censoredness, respectively. You may ignore the possibility of ties in the remainder.

• Write down the Cox proportional regression model that links overall survival times (as the response variable) to the expression levels.
• Specify its loss function for penalized maximum partial (!) likelihood estimation of the parameters. Penalization is via the regular ridge penalty.
• From this loss function, derive the estimating equation for the Cox regression coefficients.
• Describe (in words) how you would evaluate the ridge ML estimator numerically.