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In the context of Smoothing Splines, Im trying to show that the expected value of the cross-validation can approximate the predictive square error. More specifically, I want to show that

  1. $$E[(y_i - \hat{f}^{-i}_\lambda(x_i))^2] = \sigma^2 + E[(\hat{f}^{-i}_\lambda(x_i) - f(x_i))^2] $$ And that
  2. $$E[(Y_i^* - \hat{f}^{-i}_\lambda(x_i))^2] = \sigma^2 + E[(\hat{f}_\lambda(x_i) - f(x_i))^2] $$

So that if $\hat{f}^{-i}_\lambda(x_i))^2 \approx \hat{f}_\lambda(x_i))^2$ then they are approximately the same.

Note that $\hat{f}^{-i}_\lambda$ is the model fitted with smoothing parameter $\lambda$ and without the ith observation, and $Y_i^*$ is an imaginary new observation at $x_i$ obtained through the same DGP.

I can prove 1 by doing the following to the LHS and expanding out $$E[(y_i - f(x_i) + f(x_i) - \hat{f}^{-i}_\lambda(x_i))^2]$$

However I am unable to prove 2, even though I know it can be done by a similar argument. The issue is mostly that the $\hat{f}^{-i}_\lambda(x_i)$ doesnt appear in the RHS of the second equation?

Thanks!

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