Section 10f

10.7 Diagnosing Collinearity: 10.7.1 Drawbacks of Centering, 10.7.2 Detection of Points Influencing Collinearity, 10.7.3 Remedies for Collinearity

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Problem 10f.1

Show that the largest eigenvalue λ˘MAX \breve{\lambda }_{\text{MAX }} of X˘TX˘\breve{\mathbf{X}}^{T} \breve{\mathbf{X}} satisfies

1≤λ˘MAX⁡≤p 1 \leq \breve{\lambda }_{\operatorname {MAX}} \leq p
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Problem 10f.2

Show that

E[∥Xβ^−Xβ∥2]=σ2+E[∥X∗γ^−X∗γ∥2] \mathbb {E}\left[\left\| \mathbf{X} \widehat{\mathbf{\beta }}-\mathbf{X} \mathbf{\beta }\right\| ^{2}\right] = \sigma ^{2}+\mathbb {E}\left[\left\| \mathbf{X}^{*} \widehat{\mathbf{\gamma }}-\mathbf{X}^{*} \gamma \right\| ^{2}\right]
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Problem 10f.3

Prove that with the priors of Section 10.7.3 , the posterior mean of γ\gamma is just the ridge estimate (10.64).

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Problem 10f.4

Prove that with the priors of Section 10.7.3 , the prior distribution of α^\widehat{\mathbf{\alpha }} is Np−1[0,σ2(Λ−1+k−1Ip−1)]N_{p-1}\left[\mathbf{0}, \sigma^{2}\left(\mathbf{\Lambda }^{-1}+k^{-1} \mathbf{I}_{p-1}\right)\right].

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