Consider the setting of the polynomial regression in Example 2.

2. Use Theorem C.19 to prove that

\[ \begin{equation*} \sqrt{n}\left(\widehat{\boldsymbol{\beta}_{n}}-\boldsymbol{\beta}_{p}\right) \xrightarrow{\mathrm{d}} \mathscr{N}\left(0, \ell^{*} \mathbf{H}_{p}^{-1}+\mathbf{H}_{p}^{-1} \mathbf{M}_{p} \mathbf{H}_{p}^{-1}\right) \tag{2.53} \end{equation*} \]

 

where \(\mathbf{M}_{p}:=\mathbb{E}\left[\boldsymbol{X} \boldsymbol{X}^{\top}\left(g^{*}(\boldsymbol{X})-g^{\boldsymbol{Y}_{p}}(\boldsymbol{X})\right)^{2}\right]\) is the matrix with (i, \(j\) )-th entry:

\[ \int_{0}^{1} u^{i+j-2}\left(h^{\mathscr{H}_{p}}(u)-h^{*}(u)\right)^{2} \mathrm{~d} u \]

and \(\boldsymbol{H}_{p}^{-1}\) is the \(p \times p\) inverse Hilbert matrix with \((i, j)\)-th entry:

\[ (-1)^{i+j}(i+j-1)\binom{p+i-1}{p-j}\binom{p+j-1}{p-i}\binom{i+j-2}{i-1}^{2} \]

Observe that \(\mathbf{M}_{p}=\mathbf{0}\) for \(p \geqslant 4\), so that the matrix \(\mathbf{M}_{p}\) term is due to choosing a restrictive class \(\mathscr{G}_{p}\) that does not contain the true prediction function.