STAT 714, FALL 2008 HOMEWORK 3
1. Consider the matrix
A=
1 0 −1
0 1 −1
−1−1 3
.
(a) Show that Ais pd.
(b) Compute A1/2, the symmetric square root of A. Check your work by showing that
A1/2A1/2=A.
2. Suppose that yn×1and xk×1are random vectors. Define z=y−E(y|x). Show that
zand xare uncorrelated.
3. Suppose that yand xare random vectors with means µYand µX, respectively,
variance matrices ΣYand ΣX, respectively, and covariance matrix ΣY X . Assume that
ΣXis nonsingular. Define
w=µY+ΣY X Σ−1
X(x−µX)
and z=y−w. Derive cov(z) and show that cov(z)≤p d ΣY, with equality when
ΣY X =0.
4. Consider the mixed-effects linear model
yN×1=Xb +Z1e1+e2,
where Xis N×p,bis p×1, e1has mean vector 0r×1and variance-covariance matrix
Σ1, and e2has mean vector 0N×1and variance-covariance matrix σ2IN. Also, assume
that e1and e2are uncorrelated.
(a) Compute cov(y).
(b) (↑) Specialize to the one-factor random-effects model
yij =µ+αi+ij,
for i= 1,2,3 and j= 1,2, where α1, α2, α3are iid N(0, σ2
α), ij are iid N(0, σ2), and
the αi’s and ij’s are mutually independent. Put this model into the form yN×1=
Xb +Z1e1+e2, and compute cov(y).
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