Consider again the polynomial regression Example 2.

1. Use the fact that \(\mathbb{E}_{\mathbf{X}} \widehat{\boldsymbol{\beta}}=\mathbf{X}^{+} \boldsymbol{h}^{*}(\boldsymbol{u})\), where \(\boldsymbol{h}^{*}(\boldsymbol{u})=\mathbb{E}[\boldsymbol{Y} \mid \boldsymbol{U}=\boldsymbol{u}]=\left[h^{*}\left(u_{1}\right), \ldots, h^{*}\left(u_{n}\right)\right]^{\top}\), to show that the expected in-sample risk is:

\[ \mathbb{E}_{\mathbf{X}} \ell_{\text {in }}\left(g_{\mathscr{T}}\right)=\ell^{*}+\frac{\boldsymbol{h}^{*}(\boldsymbol{u})^{2}-\mathbf{x} \mathbf{X}^{+} \boldsymbol{h}^{*}(\boldsymbol{u})^{2}}{n}+\frac{\ell^{*} p}{n} \]

Also, use Theorem C. 2 to show that the expected statistical error is:

\[ \mathbb{E}_{\mathbf{X}}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})^{\top} \mathbf{H}_{p}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})=\ell^{*} \operatorname{tr}\left(\mathbf{X}^{+}\left(\mathbf{X}^{+}\right)^{\top} \mathbf{H}_{p}\right)+\left(\mathbf{X}^{+} \boldsymbol{h}^{*}(\boldsymbol{\mu})-\boldsymbol{\beta}\right) \]