Ex. 6.7
Ex. 6.7
Derive an expression for the leave-one-out cross-validated residual sum-of-squares for local polynomial regression.
Soln. 6.7
Note that local regression smoothers are linear estimators, and we can write
\[\begin{equation}
\hat{\mathbf{f}} = \bb{S}_\lambda \bb{y} \non
\end{equation}\]
where \(\{\bb{S}_\lambda\}_{ij} = l_i(x_j)\) for \(l_i(x)\) defined by (6.8) in the text. Then by Ex. 7.3 we know
\[\begin{equation}
y_i - \hat f^{-i}(x_i) = \frac{y_i - \hat f(x_i)}{1-\{\bb{S}_\lambda\}_{ii}}.\non
\end{equation}\]