 Covariance formula for prediction optimism

With fixed covariates and independent new responses having the same marginal distributions as the training responses, expanding squared losses cancels their second moments. Independence makes $\mathbb{E}(Y_i^{\mathrm{new}}{\hat{Y}}_i)=\mathbb{E}{Y}_i\mathbb{E}{\hat{Y}}_i$, yielding the displayed identity. For a fixed linear smoother $\hat{Y}=HY$ with covariance ${\sigma }^{2}I$, it reduces to $2{\sigma }^2\mathrm{tr}(H)/n$, linking prediction optimism to effective degrees of freedom.

// Target: analysis.bigb