Ex. 7.5

Ex. 7.5

For a linear smoother \(\hat{\mathbf{y}} = \bb{S}\bb{y}\), show that

\[\begin{equation} \sum_{i=1}^N\text{Cov}(\hat y_i, y_i) = \text{trace}(\bb{S})\sigma_\epsilon^2,\non \end{equation}\]

which justifies its use as the effective number of parameters.

Soln. 7.5

\[\begin{eqnarray} \sum_{i=1}^N\text{Cov}(\hat y_i, y_i) &=& \text{trace}(\text{Cov}(\hat{\mathbf{y}}, \bb{y}))\non\\ &=&\text{trace}(\text{Cov}(\bb{S}\bb{y}, \bb{y}))\non\\ &=&\text{trace}(\bb{S}\text{Cov}(\bb{y}, \bb{y}))\non\\ &=&\text{trace}(\bb{S}\text{Var}(\bb{y}))\non\\ &=&\text{trace}(\bb{S})\sigma_\epsilon^2.\non \end{eqnarray}\]

Remark

This exercise is similar to Ex. 7.1.