[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]

This question is freely copied from : Problem 2.5a, page 43.

Consider the linear regression model [math]\mathbf{Y} = \mathbf{X} \bbeta + \vvarepsilon[/math]. It is fitted to data from a study with an orthonormal design matrix by means of the adaptive lasso regression estimator initiated by the OLS/ML regression estimator. Show that the [math]j[/math]-th element of the resulting adaptive lasso regression estimator equals:

[[math]] \begin{eqnarray*} \hat{\beta}_j^{\mbox{{\tiny adapt}}} (\lambda_1) & = & \mbox{sign}(\hat{\beta}_j^{\mbox{{\tiny ols}}}) ( | \hat{\beta}_j^{\mbox{{\tiny ols}}} | - \tfrac{1}{2} \lambda_1 / | \hat{\beta}_j^{\mbox{{\tiny ols}}} |)_+. \end{eqnarray*} [[/math]]