In the book Introduction to Statistical Learning , Cp is introduced as a statistic to chose an optimal model for a linear regression setting(Page 211), Cp which is given by
it is given that if is the unbiased estimator of (ie. the variance of irreducible error) then it is given that the above statistic is an unbiased estimate for test MSE.
So my questions are,
1.What exactly is the test MSE here? Is it the Expected Prediction error? that is given by
1.a.If the first answer is yes , then how do I go about proving it is an unbiased estimator?
This is what I have tried assuming the first answer is yes, I considered the case of simple linear regression so that d=2 in the formula and finding its expected value ie.
Then the equation for Expected Prediction Error
As model assumption is unbiased, the Bias term goes to zero and we get
I am not sure how to deal with the Var(f(x)) term , heres where I am stuck any help would be appreciated.
2.If first answer is a no , then what is the test MSE authors are talking about?