2
$\begingroup$

Are Studentized deleted residuals a form of k-fold cross validation when K=N? (this question is asked in the context of the discussion here)

$\endgroup$
4
$\begingroup$

A deleted residual is $d_i = Y_i - \hat{Y}_{i(i)}$, which can be shown to be algebraically equivalent to $$ d_i = \dfrac{e_i}{1-h_{ii}}$$ where $e_i$ is the normal residual $Y_i - \hat{Y}_i$ and $h_{ii}$ is the 'hat' matrix $h_{ii} = \mathbf{X}_i^\intercal(\mathbf{X}^\intercal\mathbf{X})^{-1}\mathbf{X}_i$.

Studentizing typically means dividing the residual by its standard error. It can be shown that $$\text{Var}\left[d_i\right] = MSE_{(i)}(1+\mathbf{X}_i^\intercal(\mathbf{X}^\intercal\mathbf{X})^{-1}\mathbf{X}_i) = \dfrac{MSE_{(i)}}{1-h_{ii}}$$

The term you were referring to in the link would be $$ \dfrac{1}{N}\sum_{i=1}^N L(y_i,\hat{y}_{i(i)}) =\dfrac{1}{N}\sum_{i=1}^N d_i^2 $$ if $L$ is squared error loss

So the concepts are related, but perhaps not exactly the same. I should add that a lot of this is under the standard linear model framework.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.