[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 the standard linear regression model [math]Y_i = \mathbf{X}_{i,\ast} \bbeta + \varepsilon_i[/math] for [math]i=1, \ldots, n[/math] and with [math]\varepsilon_i \sim_{i.i.d.} \mathcal{N}(0, \sigma^2)[/math] Consider the following two ridge regression estimators of the regression parameter of this model, defined as:

[[math]] \begin{eqnarray*} \arg \min_\bbeta \sum\nolimits_{i=1}^n (Y_{i} - \mathbf{X}_{i,\ast} \bbeta)^2 + \lambda \| \bbeta \|_ 2^2 \quad \mbox{ and } \quad \arg \min_\bbeta \sum\nolimits_{i=1}^n (Y_{i} - \mathbf{X}_{i,\ast} \bbeta)^2 + n \lambda \| \bbeta \|_ 2^2. \end{eqnarray*} [[/math]]

Which do you prefer? Motivate.