Applied Statistics and Probability for Engineers

9-2

and the values of  0 and  1 are  0  0 and  1 {:    }, respectively. The values

of and 

2 that maximize Lin  1 are the usual maximum likelihood estimates for and 

2 :

Substituting these values in L, we have

To maximize Lin  0 we simply set  0 and then find the value of 

2 that maximizes L.

This value is found to be

which gives

The likelihood ratio is

Now since

we may write as

(^) c
1
1 a
1
n
1
bc
1 x
 022
s^2 n
d
s
n 2
(^) d
1
1 c
n 1 x
 022
g 1 xi x 22
da
n
1
n
1
b
t
n 2
 (^) d
1
1 
n 1 x
 022
g 1 xi x 22
t
n 2
a
n
i 1
1 xi
 (^022) a
n
i 1
1 xi x 22 n 1 x
 022

L 1  02
L 1  12
c
g 1 xi x 22
g 1 xi
 022
d
n 2
L 1  02 c
1
12 n 2 g 1 xi
 022
d
n 2
e^1 n^22
ˆ^2
1
n^ a
n
i 1
1 xi
 022
L 1  12 c
1
12 n 2 g 1 xi x 22
d
n 2
e^1 n^22
ˆ^2
1
n^ a
n
i 1
1 xi x 22
ˆ
1
n^ a
n
i 1
xi x
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