The Weibull Distribution: A Handbook - Index of

444 11 Parameter estimation — Maximum likelihood approaches
where
© 2009 by Taylor & Francis Group, LLC
g = −2 ln α (11.53b)
is the quantile of the χ 2 –distribution with two degrees of freedom and 1 − α is the level of confidence.
Introducing
C1 := r 1
+
n n
C2 := r
n ,
C3 := 1
n
�c :=
r�
i=1
c − �c
c
r�
i=1
�bc ln
�b 2
��xi � �
bc
�b
� xi
� �bc xi
ln
�b ,
��xi � �
bc
� b
�
+
�
+
1 − r
n
1 − r
n
� � �bc xr
ln
�b �� �bc xr
ln
�b 2
��xr � �
bc
,
�b ��xr � �
bc
,
�b KAHLE (1996b, p. 34) suggested the following method of constructing the confidence region:
1. Upper and lower bounds for c are
with
�c
1 + B
B =
�
≤ c ≤ �c
1 − B
C2 g
n (C1 C2 − C 2 3 ).
�
�c
2. For every c in
1 + B ,
�
�c
, the parameter b is varying in
1 − B
with
� b c
c + C3
�c + A
C2
A =
�
g
n C2
≤ b ≤
� bc
c + C3
�c − A
C2
− �c 2 C1 C2 − C2 3
C2 .
2
(11.53c)
(11.53d)
Fig. 11/6 shows the 90%–confidence region for b and c belonging to dataset #1 censored at r = 15.
Finite sample results for testing hypotheses concerningb or c or for constructing confidence intervals
for b and c rest upon the distribution of pivotal functions (cf. Sect. 11.3.2.4). BILLMAN et al. (1972)
obtained the percentage points in Tables 11/7 and 11/8 by simulation.