# Proving Mallows Cp is an unbiased estimator of test MSE

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?

• In statistics, MSE means mean square error. If I were you, I would try to find the explicit definition of test MSE in that book. If could not find, the book would go to garbage can and I would try to find another good book. – user158565 May 21 '17 at 20:05
• Yes, that would be a great idea . – GeneX May 22 '17 at 12:22