STAT 714, FALL 2008 HOMEWORK 5
1. Consider our general linear model y=Xb +e, where E(e) = 0and cov(e) = σ2I.
Let PXdenote the perpendicular projection matrix onto C(X). Let b
ydenote the vector
of least squares fitted values and b
edenote the vector of least squares residuals. Compute
each of the following.
(a) E(b
y)
(b) cov(b
y)
(c) E(b
e)
(d) cov(b
e)
(e) cov(b
y,b
e).
2. The observed tension, y, in a nonextensible string required to maintain a body of
unknown weight, w, in equilibrium on a smooth inclined plane of angle θ, 0 < θ < π/2,
is a random variable with mean E(y) = wsin θ. For Nknown values θ1, θ2, ..., θN, set by
the experimenter and a given body, the observed data are y1, y2, ..., yN.
(a). Find b
w, the least squares estimator of w, the weight of this body.
(b) Compute E(b
w) and var( b
w). You may assume that y1, y2, ..., yNare independent.
(c) Let b
y1,b
y2, ..., b
yNdenote the least squares fitted values. Is it necessarily true that
PN
i=1(yi−b
yi) = 0? Explain.
3. Define the matrices
X=
1100
1100
1010
1001
and W=
1 1
1 1
1 0
1 0
(a) Show that C(W)⊂ C(X).
(b) Take y= (1,0,1,2)0. Express yas the sum of three vectors: one in C(W), one in
C(W)⊥
C(X), and one in another space. In what space does the 3rd vector lie?
4. (a) For the simple linear regression model in Example 2.1 (notes, pp 57-59), show
algebraically that PXand PWare both equal to
1
N+(x1−x)2
Pi(xi−x)2
1
N+(x1−x)(x2−x)
Pi(xi−x)2· · · 1
N+(x1−x)(xN−x)
Pi(xi−x)2
1
N+(x1−x)(x2−x)
Pi(xi−x)2
1
N+(x2−x)2
Pi(xi−x)2· · · 1
N+(x2−x)(xN−x)
Pi(xi−x)2
.
.
..
.
.....
.
.
1
N+(x1−x)(xN−x)
Pi(xi−x)2
1
N+(x2−x)(xN−x)
Pi(xi−x)2· · · 1
N+(xN−x)2
Pi(xi−x)2.
N×N
.
In regression analysis, this matrix is called the hat matrix.
(b) Compute the trace of this matrix.
(c) Use the ceramic data from HW 4 and fit both the centred and uncentred simple linear
regression models. Report least squares estimates for both models. Also, show that the
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