Stat 511 HW#1 Spring 2003
1. Christensen Exercise 1.1 c) and d), page 3.
2. Christensen Exercise 1.11, pages 12-13.
3. Christensen Exercise 1.5.2 a)-g), page15.
4. Consider Koehler's Example 3.6 and the matrix
4 1 2 0 
A =  1 1 5 15


 3 1 3 5 
Verify (use R to do the matrix multiplication) that both
0
0 
0
 .25
0 −1.5 2.5 
 0
 and G2 = 
G1 = 
0 .5 −.5
 0



0
0 
0
 −.15
are generalized inverses for A .
0 0
0 0

0 0

0 .20
5. Christensen Exercise 1.5.8 b) page 17. Do this two ways. First, use Theorem B.33.
(You’ll need to reason that every column of X can be written as a linear combination of
columns of M and vice versa, so that C ( X ) = C ( M ) .) Second, use the construction
given in class for PX involving a generalized inverse. To find a generalized inverse for
X ′X you may use R and the result on slide 168 of Koehler’s notes (or Theorem B.19,
Corollary B.20 and the proof of Theorem B.38 in the textbook). That is, if A is a k × k
square matrix of rank r , it is possible to write A = UDV ′ where all of U , D, and V are
square, U and V have orthonormal columns (i.e., orthogonal columns, each of which has
norm 1) and D is diagonal with its first r entries nonzero. The R function svd()will
return U , V and a vector of length k giving the diagonal entries of D . If D* is diagonal,
where each non-zero element of D has been replaced by its reciprocal, then the matrix
VD*U ′
is a generalized inverse for A . (See slides 92 through 94 of Koehler’s notes for an
illustration of how to use the R function.)
6. In the context of Christensen’s Example 1.0.2, suppose that Y ′ = (2,1,3,17,10,12) .
Find Yˆ , the ordinary least squares estimate of EY = X β . Show that there is no sensible
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way to identify an “ordinary least squares estimate of β ,” by finding two different
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vectors b with Xb = Yˆ .
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