Page 356 - Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 339
^
at ^ and . Elementary calculations show that
2
q Q
ˆ 2n > 0;
q^ 2
and

n
X 2
D ˆ 4n x i x† > 0
iˆ1
The proof of this theorem is thus complete. Note that D would be zero if all
x i take the same value. Hence, at least two distinct x i values are needed for the
^
determination of ^ and .
It is instructive at this point to restate the foregoing results by using a more
compact vector–matrix notation. As we will see, results in vector–matrix form
facilitate calculations. Also, they permit easy generalizations when we consider
more general regression models.
In terms of observed sample values (x 1 , y 1 ), (x 2 , y 2 ), ..., (x n , y n ), we have a
system of observed regression equations

y i ˆ ‡ x i ‡ e i ; i ˆ 1; 2; ... ; n: 11:11†

Let

2 3 2 3 2 3
1 x 1 y 1 e 1
6 1 7 6 y 2 7 6 7
. . 7; . 7; . 7;
6 x 2 7 6 7 6 e 2 7
.
C ˆ 6 y ˆ 6 e ˆ 6
4 . . . 5 4 . . 5 4 . . 5
1 x n y n e n
and let

q ˆ :

Equations (11.11) can be represented by the vector–matrix equation

y ˆ Cq ‡ e: 11:12†

The sum of squared residuals given by Equation (11.6) is now

T
T
Q ˆ e e ˆ y Cq† y Cq†: 11:13†

TLFeBOOK