The Weibull Distribution: A Handbook - Index of

640 21 WEIBULL parameter testing
SCHAFER/SHEFFIELD (1976) did not use the MLEs �c1 and �c2 in the first and second
sample, respectively, but they used a pooled estimator ��c which results as the solution of
⎧ n�
⎪⎨ X
2n i=1
− n
��c ⎪⎩
bc n�
1i ln X1i X
i=1
n�
+
bc ⎫
2i ln X2i ⎪⎬ n� n�
n�
+ lnX1i + ln X2i = 0. (21.23a)
⎪⎭ i=1 i=1
X
i=1
bc 1i
© 2009 by Taylor & Francis Group, LLC
X
i=1
bc 2i
� �c is taken instead of (�c1 + �c2) � 2 in the above formulas where we also have to substitute � b1
and � b2 by
�� b1 =
� n�
i=1
X bc 1i
�
�
� 1 bc
n
and � � b2 =
� n�
i=1
X bc 2i
�
n
�
�1 bc
, (21.23b)
respectively. Of course, the percentage points in Tab. 21/4 do not apply to the test statistic
�
� �
�c ln�b1 + ln � �
�b2 .
SCHAFER/SHEFFIELD (1976) provided a table of the matching percentiles. The
SCHAFER/SHEFFIELD approach dominates the THOMAN/BAIN approach insofar as it has
a little more power.
Readers interested in a k–sample test of H0: b1 = b2 = ... = bk against HA: bi �= bj for
at least one pair of indices (i,j), i �= j, are referred to
• MCCOOL (1977) who develops a procedure in analogy to the one–way analysis of
variance using certain ratios of MLEs of the common but unknown shape parameters;
• CHAUDHURI/CHANDRA (1989) who also present an ANOVA test, but based on sample
quantiles;
• PAUL/THIAGARAJAH (1992) who compare the performance of several test statistics.
In all three papers the k WEIBULL populations are assumed to have a common but unknown
shape parameter.
Readers who prefer GLUEs over MLEs are referred to BAIN (1978), ENGELHARDT/BAIN
(1979) and BAIN/ENGELHARDT (1991a) where the one–sample test and the k–sample test
(k ≥ 2) are treated.
21.1.3 Hypotheses concerning the location parameter a 9
Perhaps the most interesting hypothesis concerning the location parameter a is whether
a = 0. MCCOOL (1998) has suggested an elegant procedure to test H0 : a = 0 versus
9 Suggested reading for this section: DUBEY (1966a), MCCOOL (1998), SCHAFER (1975).