# Expected Value of Cross Validation approximates Predictive Square Error

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!