Page 264 - Elements of Distribution Theory

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052184472Xc08 CUNY148/Severini May 24, 2005 17:54

250 Normal Distribution Theory

Example 8.11 (Analysis of variance). Let X denote a d-dimensional random vector with
a multivariate normal distribution with mean vector µ and covariance matrix given by I d .
d
Let M denote a p-dimensional linear subspace of R and let P M be the matrix representing
orthogonal projection onto M; here orthogonality is with respect to the usual inner product
d
d
on R . Hence, for any x ∈ R , P M x ∈ M and
T
(x − P M x) y = 0 for all y ∈ M.
Note that P M has rank r, the dimension of M. Consider the linear transformation given
by the matrix I d − P M .Itis easy to show that this matrix represents orthogonal projection
onto the orthogonal complement of M; hence, the rank of I d − P M is d − r.
Since
T
T
T
X X = X (I d − P M )X + X P M X,
T
T
it follows that the quadratic forms X (I d − P M )X and X P M X are independent chi-
squared random variables with d − r and r degrees of freedom, respectively.
d
Now suppose that R may be written
d
R = M 1 ⊕ M 2 ⊕· · · ⊕ M J
d
where M 1 , M 2 ,..., M J are orthogonal linear subspaces of R so that if x i ∈ M i and
x j ∈ M j , i = j,
T
x x j = 0.
i
denote orthogonal projection onto M j and let r j denote the dimension of M j ,
Let P M j
j = 1,..., J;it follows that r 1 +· · · + r J = d.
T
Let Q j = X P M j X, j = 1,..., J. Then
T
X X = Q 1 +· · · + Q J
and Q 1 ,..., Q J are independent chi-squared random variables such that Q j has degrees
of freedom r j , j = 1,..., J.

8.5 Sampling Distributions
In statistics, the results of this chapter are often applied to the case of independent real-
valued, normally distributed random variables. In this section, we present some classic
results in this area.

Theorem 8.9. Let X 1 ,..., X n denote independent, identically distributed standard normal
random variables. Let
n
1
¯
X = X j
n
j=1
and let
n
1
2 ¯ 2
S = (X j − X) .
n − 1
j=1