0
$\begingroup$

As posted here in the book Elements of Statistical Learning we drive to the following: $$ w = \frac{2}{N}\sum_1^Ncov(y_i, \hat{y_i}) $$

The derivation is based on the following properties

  1. $E_y E_{Y^0}[(Y_i^0)^2] = E_y[y_i^2]$
  2. $2 E_y E_{Y^0} [Y_i^0 \hat{y}_i]=2 E_y [E_{Y^0} [Y_i^0]E_{Y^0}[\hat{y}_i]]=2 E_y [y_i] E_y[ \hat{y}_i]$

My question is how exactly do we prove these properties? As far as I've figured out for the first statement we use tower property of conditional expectation. For the second one we use independence of $Y_i^0$ and $\hat{y_i}$.

Is this fair or shall I look into some other explanation?

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.