Page 270 - Elements of Distribution Theory

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256 Normal Distribution Theory

d
where ||·|| denotes the Euclidean norm in R , that is,
2
T
||x|| = x x.
(a) Find the distribution of D(M).
d
(b) Let M 1 and M 2 denote linear subspaces of R . Find conditions on M 1 and M 2 under
which D(M 1 ) and D(M 2 ) are independent.
8.24 Let X denote an n-dimensional random vector with a multivariate normal distribution with
2
mean vector 0 and covariance matrix given by σ I n . Let M 1 and M 2 denote orthogonal linear
n
subspaces of R and let P j denote the matrix representing orthogonal projection onto M j ,
j = 1, 2. Find the distribution of
T
X P 1 X
.
T
X P 2 X
8.25 Prove Corollary 8.1.

8.7 Suggestions for Further Reading
An excellent reference for properties of the multivariate normal distribution and the associated sam-
pling distributions is Rao (1973, Chapter 3). Stuart and Ord (1994, Chapters 15 and 16) contains a
detailed discussion of the distribution theory of quadratic forms and distributions related to the normal
distribution, such as the chi-squared distribution and the F-distribution. Many books on multivariate
statistical analysis consider the multivariate normal distribution in detail; see, for example, Anderson
(1984) and Johnson and Wichern (2002).