Page 258 - Elements of Distribution Theory

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244 Normal Distribution Theory

is a generalized inverse of 22 . Furthermore,
†
22 † = †
22 22 22
†
†
−
and both 22 and 22 are symmetric. It follows that † = , the Moore–Penrose
22 22 22 22
inverse of 22 .
In fact, it has been shown by Rao (1973, Section 8a.2) that the result of Theorem 8.4
holds for any choice of generalized inverse.
Example 8.8 (Bivariate normal distribution). Suppose that X has a bivariate normal
distribution, as discussed in Example 8.5, and consider the conditional distribution of
X 1 given X 2 = x 2 . Suppose that the covariance matrix of X 2 is singular; that is, sup-
pose that σ 2 = 0. The Moore–Penrose generalized inverse of 0 is 0 so that the conditional
2
distribution of X 1 given X 2 = x 2 is a normal distribution with mean µ 1 and variance σ .
1
T
This holds provided that x 2 is such that for any vector a such that a X 2 has variance 0,
T
T
a x 2 = a µ 2 ;in this case, this means that we require that x 2 = µ 2 . Note that X 2 = µ 2 with
probability 1.
8.4 Quadratic Forms
Much of this chapter has focused on the properties of linear functions of a multivariate
normal random vector X;however, quadratic functions of X also often occur in statistical
T
methodology. In this section, we consider functions of X of the form X AX where A is a
symmetric matrix of constants; such a function is called a quadratic form.We will focus
on the case in which A is a nonnegative-definite matrix.

Example 8.9 (Sample variance). Let X 1 ,..., X n denote independent, identically dis-
tributed, real-valued random variables such that X j has a normal distribution with mean 0
2
and variance σ and let
n
1
2 ¯ 2
S = (X j − X)
n − 1
j=1
¯ n
where X = X j /n.
j=1
¯
Let X = (X 1 ,..., X n ).Then X = mX wherem denotesa1 × n vectorwitheachelement
n
taken to be 1/n. Since, for any vector c = (c 1 ,..., c n ) ∈ R ,
n

2 T
c = c c,
j
j=1
n

T T T T T T T
¯ 2
(X j − X) = (X − nm mX) (X − nm mX) = X (I n − nm m) (I n − nm m)X.
j=1
Let
1 T T T 1 T
A = (I n − nm m) (I n − nm m) = (I n − nm m);
n − 1 n − 1
2
T
then S = X AX.