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Proof of Distribution
The goal here is to show that Equation 3.13.21, that is
has an distribution. We will do this by showing that the numerator
and the denominator are both times chisquare variables
divided by their degrees of freedom.
Since
is a linear function of Y, and
, a vector of iid
random variables, it follows from
Equations 3.13.7 and 3.13.8 that
is a vector of random variables with a
distribution. Thus, the transformation,
, results in
random variables ( being the number of rows in C, the
number of tests):
Under the hypothesis that
we have,
thus,
Since the sum of squared iid variables results in a random
variable distributed by , it follows from
Equation D.7.1 that,
Thus, we have shown that the numerator in Equation 3.13.21 is
times a chisquare random variable divided by its degrees
of freedom.
Showing the same thing for the denominator is a little more tricky as
it involves some obscure transformations and knowing a few properties
of quadratic forms. Instead of trying to explain the details about
quadratic forms that would be required for a full proof, we'll simply
go as far as we can with what we have and appeal to your sense of intuition.
Let
,
an approximation of the error vector,
,
thus,
and







(D.7.2) 



(D.7.3) 



(D.7.4) 



(D.7.5) 
Equations D.7.2, D.7.3, D.7.4 and D.7.5
make it clear that
is idempotent, that is,
We can determine the rank^{D.1} of
from its trace
(that is, the sum the elements on the diagonal) since it is idempotent and symmetric. Using a well known
property of traces, that is, tr tr, and the fact
that X is an
matrix, we have
Since
, it follows that
, thus we can imagine that
Equation D.7.5 is the sum of independent
normal random variables. Thus, if we let
, then
Showing that the numerator is independent from the denominator
also requires some obscure transformations and requires another result
from quadratic forms. Without proof, I will state that the following theorem.
Let Z is a vector of of normally distributed random
variables with a common variance and let
and
, where A and B are both
symmetric matrices. and are independently
distributed if and only if
.
Now, to show independence, under the hypothesis that C
, can rewrite the ends
of the numerator with
If we let
, then we can rewrite the numerator of
Equation 3.13.21 as
. If we let
, then we can rewrite the
denominator to be
. Since
, and thus, under the hypothesis, the numerator and
denominator are independent.
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Frank Starmer
20040519