The linear model ${\bf Y}={X\beta}+\epsilon$, where ${\bf Y}$ is a $n\times 1$ vector, and ${\bf X}$ is $n\times p$ matrix. $n\lt p$ and $rank({\bf X})=n$. $\epsilon\sim N(0, \sigma^2)$. How to prove $\|X(X^TX)^{-1}X^TY-Y+\epsilon\|_2^2/\sigma^2$ is $\chi^2_p$-distributed.
I think this statement in $\textit{Statistics for High Dimensional Data: Methods, Theory and Applications}$ by P. Buhlmann and S. Van de Geer is wrong. $\|X(X^TX)^{-1}X^T-Y+\epsilon\|_2^2/\sigma^2$ should be $\chi^2_n$-distributed.