14.5The Weibull Distribution in Life Testing 601
Weibull (1, 0.5)
x
012345
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Weibull (1, 2)
x
012345
1.0
0.8
0.6
0.4
0.2
0.0
Weibull (1, 1)
x
012345
1.0
0.8
0.6
0.4
0.2
0.0
Weibull (1, 3)
x
012345
1.2
1.0
0.8
0.6
0.4
0.2
0.0
FIGURE 14.2 Weibull density functions.
and
∂
∂α
logf(x 1 ,...,xn)=
n
α
−
∑n
i= 1
xβi
∂
∂β
logf(x 1 ,...,xn)=
n
β
+
∑n
i= 1
logxi−α
∑n
i= 1
xiβlogxi
Equating to zero shows that the maximum likelihood estimatesαˆandβˆare the solutions of
n
αˆ
=
∑n
i= 1
x
βˆ
i
n
βˆ
+
∑n
i= 1
logxi=ˆα
∑n
i= 1
x
βˆ
i logxi