The Weibull Distribution: A Handbook - Index of

6.5 Characterizations of related WEIBULL distributions 271
Theorem 24 (PAWLAS/SZYNAL, 2000a)
Fix a positive integer k ≥ 1 and let r be a non–negative integer. A necessary and sufficient
condition for X to be distributed with DF
is that
�� E
Y (k)
m
� r+c �
© 2009 by Taylor & Francis Group, LLC
f(x) = λcx c−1 exp � − λx c� ; x ≥ 0; c,λ > 0 (6.34b)
�� = E Y (k)
� �
r+c
m−1 + r + c
k λc E
��
Y (k)
m
� r �
for m = 1,2,... �
(6.34c)
ROY/MUKHERJEE (1986) gave several WEIBULL characterizations. One of them departs
from the hazard rate
h(x) = f(x)
1 − F(x) ,
but takes the argument to be a random variable X so that h(X) is a variate too. The
characterization pertains to the distribution of h(X).
Theorem 25 (ROY/MUKHERJEE, 1986)
• h(·) is strictly increasing with h(0) = 0.
• h(X) is WEIBULL with shape parameter c ′ > 1 and scale factor λ ′ , for the
parametrization see (6.27a), iff X is WEIBULL with shape parameter c > 1, (1/c +
1/c ′ ) = 1, and λ ′ = λ(cλ ′ ) c . �
They further showed that the FISHER–information is minimized among all members of a
class of DF’s with
• f(x) continuously differentiable on (0, ∞),
• xf(x) → 0 as x → 0 + , x 1+c f(x) → 0 as x → ∞,
• � x c f(x)dx = 1 and � x 2 c f(x)dx = 2,
when f(x) is the WEIBULL distribution with scale factor equal to 1 and c as its shape
parameter. 6 Another characterization theorem given by ROY/MUKHERJEE (1986) says
that the maximum entropy distribution — under certain assumptions — is the WEIBULL
distribution.
6.5 Characterizations of related WEIBULL distributions
In Sect. 3.3.6.5 we have presented several compound WEIBULL distributions, one of them
being the WEIBULL–gamma distribution (see (3.73c,d)), where the compounding distribution
is a gamma distribution for B in the WEIBULL parametrization
Here, we take the parametrization
f(x |B) = cB (x − a) c−1 exp � − B (x − a) c� .
f(x |β) = cβ −1 (x − a) c−1 exp � − β −1 (x − a) c�
(6.35a)
6 GERTSBAKH/KAGAN (1999) also characterized the WEIBULL distribution by properties of the FISHER–
information but based on type–I censoring.