12.9 Comparison of Methods: 12.9.1 Identifying the Correct Subset, 12.9.2 Using Prediction Error as a Criterion

Show that the function $f$ which minimizes the expected ME (12.80) is the conditional expectation $\mathbb {E}\left[\beta \mid \widehat{\beta }\right]$ given by (12.81).

Draw the same diagram for Example 12.6 as those given in the text in Examples 12.4 and 12.5.

Show that in ridge regression when $\mathbf{X}^{T} \mathbf{X} = \mathbf{I}_{p}$, the little bootstrap and GCV give the same estimate for the ridge parameter.