distribution of $\|P_VX\|^2$ with orthogonal projection $P$ onto $V$

Question

We've had the following question discussed today but without any result:

Let $X_1,\dots,X_d$ be random variables, iid and $X_n\sim N(\mu_n,1)$.

How can we describe the distribution of $\|P_VX\|^2$ with $X=(X_1,\dots,X_d)$, $V\subset \mathbb R^d$ and an orthogonal projection $P$?

mookid · Accepted Answer · 2014-10-29 21:18:08Z

0

Hint: use the fact that $$ P_V(I - P_V) = 0 $$ and the Cochran theorem.

answered Oct 29 '14 at 21:18

mookid

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$\begingroup$ That is the standard one, at least. This is a very classical approach presented in many books of statistics. $\endgroup$ – mookid Oct 29 '14 at 23:07
$\begingroup$ My problem is to find the $q_i$'s, how do you get them? $\endgroup$ – Ele Oct 29 '14 at 23:16
$\begingroup$ The version of the theorem in the link is not the most clear I have seen, I'm sorry for that. I'll give a working version here: if $X\sim N(\mu,\Sigma)$ and if you can find orthogonal projectors $p_i$ whose sum is $id$ and such as $i\neq j\implies p_i\circ p_j = 0$, then the variables $p_i X$ are independant, with marginals $N(p_i\mu, p_i \Sigma p_i)$ $\endgroup$ – mookid Oct 29 '14 at 23:27

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