# 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 $\hat{\sigma^2}$ is the unbiased estimator of $\sigma^2$(ie. the variance of irreducible error) then it is given that the above $C_{p}$ 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

$EPE({x_{o}})&space;=&space;Var(\hat{f(x_{o})})&space;+&space;Bias^2(\hat{f(x_{o})})+\sigma^2$

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.

$E(C_{p})&space;=&space;\frac{1}{n}E(RSS+4\hat{\sigma}^2)$

$E(RSS)&space;=&space;(n-2)\sigma^2$

$E(\hat{\sigma}^2)&space;=&space;\sigma^2$

$\therefore&space;E(C_{p})&space;=&space;\frac{n+2}{n}\sigma^2$

Then the equation for Expected Prediction Error

As model assumption is unbiased, the Bias term goes to zero and we get

$EPE(x_{o})&space;=&space;Var(\hat{f(x_{o})})&space;+&space;\sigma^2$

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