Page 267 - Elements of Distribution Theory

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8.6 Exercises 253

T
T
−1
T ˆ
and c (β − β) = a P M X where a = Z(Z Z) c.Itnow follows from Theorem 8.10 that
T has a t-distribution with n − 2degrees of freedom.

8.6 Exercises
8.1 Let X denote a d-dimensional random vector with a multivariate normal distribution with covari-
ance matrix satisfying | | > 0. Write X = (X 1 ,..., X d ), where X 1 ,..., X d are real-valued,
and let ρ ij denote the correlation of X i and X j for i = j.
Let R denote the d × d matrix with each diagonal element equal to 1 and (i, j)th element equal
to ρ ij , i = j. Find a d × d matrix V such that
= VRV.

8.2 Let Y denote a d-dimensional random vector with mean vector µ. Suppose that there exists
d
T
d
T
m ∈ R such that, for any a ∈ R ,E(a Y) = a m. Show that m = µ.
8.3 Let X = (X 1 ,..., X d )havea multivariate normal distribution with mean vector µ and covari-
, the joint cumulant of
ance matrix .For arbitrary nonnegative integers i 1 ,..., i d , find κ i 1 ···i d
(X 1 ,..., X d )of order (i 1 ,..., i d ).
8.4 Let X denote a d-dimensional random vector with a multivariate normal distribution with mean
µ and covariance matrix . Let (λ 1 , e 1 ),..., (λ d , e d ) denote the eigenvalue–eigenvector pairs of
T
, λ 1 ≥ λ 2 ≥ ··· ≥ λ d .For each j = 1,..., d, let Y j = e X and let Y = (Y 1 ,..., Y d ). Find the
j
covariance matrix of Y.
8.5 Let X = (X 1 ,..., X d ) where X 1 ,..., X d are independent random variables, each normally dis-
tributed such that X j has mean µ j and standard deviation σ> 0. Let A denote a d × d matrix
of constants. Show that
T
2
T
E(X AX) = σ tr(A) + µ Aµ
where µ = (µ 1 ,...,µ d ).
8.6 Let X 1 and X 2 denote independent, d-dimensional random vectors such that X j has a multivariate
normal distribution with mean vector µ j and covariance matrix j , j = 1, 2. Let X = X 1 + X 2 .
Find the mean vector and covariance matrix of X. Does X have a multivariate normal distribution?
8.7 Consider a multivariate normal random vector X = (X 1 , X 2 ,..., X n ) and suppose that
X 1 , X 2 ,..., X n are exchangeable random variables, each with variance 1, and let denote
the covariance matrix of the distribution. Suppose that n X n = 1 with probability 1;
j=1
find .
8.8 Consider a multivariate normal random vector X = (X 1 , X 2 ,..., X n ) and suppose that
X 1 , X 2 ,..., X n are exchangeable random variables, each with variance 1, and let denote
the covariance matrix of the distribution. Then
 
1 ρρ ··· ρ
=  ρ 1 ρ ··· ρ 
ρρρ ··· 1
for some constant ρ; see Example 8.2. Find the eigenvalues of and, using these eigenvalues,
find restrictions on the value of ρ so that is a valid covariance matrix.
8.9 Let X = (X 1 , X 2 ,..., X n ) where X 1 , X 2 ,..., X n are independent, identically distributed ran-
dom variables, each with a normal distribution with mean 0 and standard deviation σ. Let B
denote an orthogonal n × n matrix and let Y = (Y 1 ,..., Y n ) = BX. Find the distribution of
Y 1 ,..., Y n .