Page 257 - Elements of Distribution Theory

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

8.3 Conditional Distributions 243

and covariance matrix

−
11 − 12 21 ,
22
T
T
T
provided that x 2 is such that for any vector a satisfying a 22 a = 0,a x 2 = a µ 2 . Here
denotes the Moore–Penrose generalized inverse of 22 .
−
22
Proof. By part (iii) of Theorem 8.1, there is a linear transformation of X 2 to (Y 1 , Y 2 ) such
that Y 1 is constant with probability 1 and Y 2 has a multivariate normal distribution with full-
rank covariance matrix. Furthermore, Y 2 = CX 2 where C is an r × (d − p) matrix with
rows taken to be the eigenvectors corresponding to nonzero eigenvalues of 22 ; see the
proof of Theorem 8.1. Hence, the conditional distribution of X 1 given X 2 is equivalent to
the conditional distribution of X 1 given Y 2 . Since the covariance matrix of Y 2 is of full-rank,
it follows from Theorem 8.3 that this conditional distribution is multivariate normal with
mean vector
−1
µ 1 + 13 (y 2 − µ 3 )
33
and covariance matrix
−1
11 − 13 31
33
where µ 3 denotes the mean of Y 2 , 13 denotes the covariance of X 1 and Y 2 , and 33 denotes
the covariance matrix of Y 2 .
By considering the transformation

I p 0
X 1 X 1
= ,
Y 2 0 C X 2
it follows from Theorem 8.1 that
T
13 = 12 C ,
T
33 = C 22 C ,
and µ 3 = Cµ 2 . Hence, the conditional distribution of X 1 given X 2 = x 2 is multivariate
normal with mean
T
T −1
µ 1 + 12 C [C 22 C ] C(x 2 − µ 2 )
and covariance matrix
T −1
T
11 − 12 C [C 22 C ] C 21 ,
T
T
T
provided that x 2 is such that for any vector a such that a X 2 has variance 0, a x 2 = a µ 2 ;
see Example 8.8 below for an illustration of this requirement.
T
Recall that 22 = C DC where D is a diagonal matrix with diagonal elements taken to
be the nonzero eigenvalues of 22 . Note that
T
T −1
T
−1
T
T
T
T
22 C [C 22 C ] C 22 = C D(CC )[(CC )D(CC )] (CC )DC
T
= C DC = 22 ,
T
since (CC ) and D are invertible. Hence,
T
T −1
† ≡ C [C 22 C ] C
22