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Currently working through Introduction to Statistical learning, and on page 76 I am struggling with the following:

They state the equation for the F statistic as:

$F = \frac{\frac{TSS − RSS}{p}} {\frac{RSS}{n−p−1}}$

where, TSS = $\sum(y_i − y)^2$ and RSS = $\sum(y_i −yˆi)^2$

Then they say:

If the linear model assumptions are correct, one can show that $E \frac{RSS}{n − p − 1}$ = $\sigma^2$ and that, provided $H_0$ is true, $E\frac{TSS − RSS}{p}$ = $\sigma^2$.

Can someone please explain this?

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  • $\begingroup$ Are you looking for proofs? or just plain English explanation? Proofs available all over the Internet. $\endgroup$ – SmallChess Oct 9 '17 at 22:07
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Your model is $y_i = \beta_0 + \beta X_i + e$ with the $e \sim_{IID}$ normal$(0, \sigma^2)$. $\beta$ is a vec of length $p$, $X$ a model matrix of dim $n \times p$.

The correct expression for TSS is $\sum(y_i - \bar{y})$ with $\bar{y}$ the sample mean.

The first expectation is a direct consequence of the model assumption, using a degrees of freedom correction.

If the null hypothesis is true, what do we know of $\hat{y}$'s value?

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