STAT611
Exam 2
November 6, 2012
1. Suppose y = Xβ + , where y is a vector of responses, X is a known and fixed n × p
design matrix, β is an unknown parameter vector in Rp , and is a random and unobserved
error vector.
(a) What additional assumptions are necessary for the Aitken Model to hold?
(b) Suppose the Aitken Model holds. Prove that w0 X 0 V −1 y is the BLUE of an estimable
function c0 β if and only if X 0 V −1 Xw = c.
iid
2. Suppose Z1 , Z2 , Z3 ∼ N (0, 1). Derive the distribution of
W =
2(Z1 + Z2 + Z3 )2 + 3(Z1 − Z2 )2 + 12(Z1 − Z2 ) + 12
.
2(Z1 + Z2 − 2Z3 )2
iid
iid
3. Suppose y1 , . . . , y10 ∼ N (−1, σ 2 ) independent of y11 , . . . , y20 ∼ N (1, σ 2 ). Find the expected value of
20
1 X
s2 =
(yi − ȳ· )2 .
19 i=1
4. Suppose A is an n × r matrix of rank r.
(a) Prove that the eigenvalues of A0 A are the same as the non-zero eigenvalues of AA0 .
P √
(b) Prove that A = ri=1 λi ui vi0 , where λ1 , . . . , λr are the eigenvalues of A0 A, v i is a
unit-norm eigenvector of A0 A corresponding to λi , and ui is a unit-norm eigenvector
of AA0 corresponding to λi for all i = 1, . . . , r.
1
5. Suppose two treatments are randomly assigned to sows (mother pigs) using a completely
randomized design. Let ni denote the number of sows receiving treatment i for i = 1, 2.
Suppose that the jth sow receiving treatment i has mij piglets (baby pigs), and let yijk denote
the weight of piglet k born to sow j in treatment group i (i = 1, 2; j = 1, . . . , ni ; k =
1, . . . , mij ). Consider the model
yijk = µi + sij + eijk ,
where µ1 and µ2 are real-valued parameters, sij ∼ N (0, σs2 ), and eijk ∼ N (0, σe2 ) for all
i = 1, 2; j = 1, . . . , ni ; k = 1, . . . , mij . Furthermore, suppose all sij and eijk terms are
mutually independent, and suppose σs2 and σe2 are positive variance components satisfying
σs2 /σe2 = 3. Let
mij
ni X
ni
X
X
yi·· =
yijk and mi· =
mij for i = 1, 2.
j=1 k=1
j=1
Fill in the blank in the statement below with a condition that is straightforward to understand
and easy to check. (Although you do not need to prove that your answer is correct, you may
wish to show your work or explain your reasoning so that you can receive partial credit if
your answer is not right.)
The BLUE of µ1 − µ2 is
y1··
y2··
−
if and only if
m1· m2·
2
.