Page 250 - Elements of Distribution Theory

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

236 Normal Distribution Theory

T
where 11 is p × p, 12 = is p × (d − p), and 22 is (d − p) × (d − p). Then
21
X 1 has a multivariate normal distribution with mean vector µ 1 and covariance
matrix 11 .
(v) Using the notation of part (iv), X 1 and X 2 are independent if and only if 12 = 0.
(vi) Let Y 1 = M 1 X and Y 2 = M 2 X where M 1 is an r × d matrix of constants and M 2
is an s × d matrix of constants. If M 1 M 2 = 0 then Y 1 and Y 2 are independent.
(vii) Let Z 1 ,..., Z d denote independent, identically distributed standard normal ran-
dom variables and let Z = (Z 1 ,..., Z d ). Then Z has a multivariate normal distri-
bution with mean vector 0 and covariance matrix given by the d × d identity matrix.
Arandom vector X has a multivariate normal distribution with mean vector µ and
1
covariance matrix if and only if X has the same distribution as µ + 2 Z.
T
T
d
Proof. Let a ∈ R . Since a X has a normal distribution with mean a µ and variance
T
a a,it follows that
2
t
T T T
E[exp{it(a X)}] = exp it(a µ) − (a a) .
2
d
Hence, for any t ∈ R ,

1
T T T
E[exp{it X}] = exp it µ − t t ,
2
proving part (i).
T
T
d
T
p
Let a ∈ R . Then B a ∈ R so that a BX has a normal distribution with mean a Bµ
T
p
T
T
and variance a B B a; that is, for all a ∈ R , a (BX) has a normal distribution with
T
T
T
mean a (Bµ) and variance a (B B )a.Part (ii) of the theorem now follows from the
definition of the multivariate normal distribution.
Suppose that has rank r; let (λ 1 , e 1 ),..., (λ r , e r ) denote the eigenvalue–eigenvector
pairs of , including multiplicities, corresponding to the nonzero eigenvalues so that
T
T
= λ 1 e 1 e +· · · + λ r e r e .
r
1
d
Consider the linear subspace of R spanned by {e 1 ,..., e r } and let V denote the orthog-
T
onal complement of that space. Then, for any v ∈ V , v = 0; hence, v X has a normal
T
distribution with mean v µ and variance 0, proving the first part of (iii). For the matrix C
T
T
take the r × d matrix with jth row given by e . Then C C is the diagonal matrix with
j
jth diagonal element λ j . This proves the second part of (iii).
Part (iv) of the theorem is a special case of part (ii) with the matrix B taken to be of the
form
B = ( I p 0)
where I p is the p × p identity matrix and 0 is a p × (d − p) matrix of zeros.
Let ϕ 1 denote the characteristic function of X 1 , let ϕ 2 denote the characteristic function
of X 2 , and let ϕ denote the characteristic function of X = (X 1 , X 2 ). Then, from parts (i) and
(iv) of the theorem,
1 T p

T
ϕ 1 (t) = exp it µ 1 − t 11 t , t ∈ R ,
2