Student Notes To Accompany MS4214: STATISTICAL INFERENCE

Relative Likelihood
0.4 0.6 0.8 1.0
Radioactive Scatter
−1.0 −0.5 0.0 0.5 1.0
θ
θ ^ = 0.2003788
Figure 2.6.2: Relative likelihood for the radioactive scatter, solved by Newton Raphson.
A Weibull random variable with ‘shape’ parameter a > 0 and ‘scale’ parameter
b > 0 has density
fT (t) = (a/b)(t/b) a−1 exp{−(t/b) a }
for t ≥ 0. The (cumulative) distribution function is
FT (t) = 1 − exp{−(t/b) a }
on t ≥ 0. Suppose that the time to failure T of components has a Weibull distribution
and after testing n components for 100 hours, m components fail at times t1, . . . , tm,
with n − m components surviving the 100 hour test. The likelihood function can be
written
L(θ) =
m�
� �a−1 � � �a� a ti
ti
exp −
b b
b
i=1
� �� �
components failed
33
n�
� � �a� 100
exp − .
b
j=m+1
� �� �
components survived