Page 373 - Fundamentals of Probability and Statistics for Engineers

356 Fundamentals of Probability and Statistics for Engineers

Answer: in this case, C is a 12 3 matrix and
;
2 3
12 626 290
T 6 7
626 36; 776 15; 336 5;
C C ˆ 4
290 15; 336 7; 028
2 3
2; 974
T 6 7
159; 011 5:
C y ˆ 4
72; 166
T
We thus have, upon finding the inverse of C C by using either matrix inversion
formulae or readily available matrix inversion computer programs,
2 3
33:84
^ T 1 T 4 0:39 ;
q ˆ C C† C y ˆ
5
10:80
or
^
^
^
0 ˆ 33:84; 1 ˆ 0:39; 3 ˆ 10:80:
The estimated regression equation based on the data is thus
^ ^ ^
Efygˆ 0 ‡ 1 x 1 ‡ 2 x 2
d
ˆ 33:84 ‡ 0:39x 1 ‡ 10:80x 2 :
Since Equation (11.48) is identical to its counterpart in the case of simple linear
regression, much of the results obtained therein concerning properties of least-
square estimators, confidence intervals, and hypotheses testing can be dupli-
cated here with, of course, due regard to the new definitions for matrix C and
vector . q
Let us write estimator Q for q in the form
^
^
T
T
1
Q ˆ C C† C Y: 11:50†
We see immediately that
^
T
1
T
EfQgˆ C C† C EfYgˆ q: 11:51†
Hence, least-square estimator Q is again unbiased. It also follows from Equa-
^
tion (11.21) that the covariance matrix for Q is given by
^
^
1
2
T
covfQgˆ C C† : 11:52†

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