Page 355 - Fundamentals of Probability and Statistics for Engineers

338 Fundamentals of Probability and Statistics for Engineers

where
n
1 X
x ˆ x i ;
n
iˆ1
and
n
1 X
y ˆ y i :
n
iˆ1
Proof of Theorem 11.1: estimates ^ and ^ are found by taking partial
^
derivatives of Q given by Equation (11.6) with respect to ^ and , setting these
,
^
derivatives to zero and solving for ^ and . Hence, we have
n
qQ X
^
ˆ 2‰y i ^ ‡ x i †Š;
q^
iˆ1
n
qQ X
^
ˆ 2x i ‰y i ^ ‡ x i †Š:
q ^ iˆ1
Upon simplifying and setting the above equations to zero, we have the so-called
normal equations:
^
n^ ‡ nx ˆ ny; 11:9†
n X
n
X
2
nx^ ‡ ^ x ˆ x i y i : 11:10†
i
iˆ1 iˆ1
Their solutions are easily found to be those given by Equations (11.7) and
(11.8).
To ensure that these solutions correspond to the minimum of the sum of
squared residuals, we need to verify that
2
q Q
> 0;
q^ 2
and

2 2
q Q
q Q

q^ 2
^
> 0;
q^ q
D ˆ 2 2
q Q q Q

q^ q q
^ ^2

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