STAT611
Exam 1
September 25, 2012
1. Consider the general linear model discussed in class, where y = Xβ + with E() = 0.
(a) State the definition of a linearly estimable function c0 β.
(b) State the definition of a least squares estimator of a linearly estimable function c0 β.
(c) State the normal equations.
(d) Prove that if c0 β̂ is the same for all solutions β̂ to the normal equations, then c0 β is
linearly estimable.
2. Suppose A is an m × n matrix. Prove that any matrix G satisfying A0 AG = A0 is a
generalized inverse of A.
3. Suppose A is an m × n matrix and B is an n × r matrix. Prove that if rank(A) = n, then
B − A− is a generalized inverse of AB.
4. Is the following statement true or false?
If S and T are vector spaces in Rn with dim(S) < dim(T ), then there always
exists a nonzero vector x ∈ T such that x ∈ S ⊥ .
(You are only required to answer either true or false.)
1
5. Suppose two fertilizers (1 and 2) were randomly assigned to six pots, with three pots for each
fertilizer. Suppose the three pots that received fertilizer 1 were randomly assigned to receive
1, 2, or 3 units of fertilizer 1. Likewise, suppose the three pots that received fertilizer 2 were
randomly assigned to receive 1, 2, or 3 units of fertilizer 2. Each pot contained a seedling,
and the height of each seedling was recorded six weeks after the application of fertilizer. For
i = 1, 2 and j = 1, 2, 3, let yij denote the height of the seedling in the jth pot that received
fertilizer i, and let xij denote the amount of fertilizer applied to the jth pot that received
fertilizer i. Suppose the data are as follows.
Pot
1
2
3
4
5
6
i
1
1
1
2
2
2
j Fertilizer
1
1
2
1
3
1
1
2
2
2
3
2
Amount of Fertilizer (xij )
1
2
3
1
2
3
Height of Seedling (yij )
8
12
14
6
15
17
Consider the general linear model given by
E(yij ) = µ + φi + γi xij + γ3 x2ij
for all i = 1, 2 and j = 1, 2, 3, where µ, φ1 , φ2 , γ1 , γ2 , and γ3 are unknown parameters.
Define β = (µ, φ1 , φ2 , γ1 , γ2 , γ3 )0 and y = (y11 , y12 , y13 , y21 , y22 , y23 )0 .
(a) Determine the design matrix X so that E(y) = Xβ.
(b) Is µ + φ1 estimable? Prove that your answer is correct.
(c) C(X) = C(W ), where


1
1 −1
0
1
1
1
0
0 −2


1

1
1
0
1
.
W =
1 −1
0 −1
1


1 −1
0
0 −2
1 −1
0
1
1
Use this fact to find β̂ such that X β̂ = P X y.
2