The Weibull Distribution: A Handbook - Index of

14.2 Two-parameter WEIBULL distribution 521
Introducing y = λT , the denominator of (14.17b) becomes
�B
A
© 2009 by Taylor & Francis Group, LLC
λ r exp(−λT) dλ =
=
�B
T
A T
y r exp(−y)
T r+1
dy
1
T r+1
� �
γ(r + 1 |B T) − γ(r + 1 |AT) , (14.17c)
where γ(· |·) represents the incomplete gamma function (see the excursus on the gamma
function in Sect. 2.9). Substituting (14.17c) into (14.17b) we finally have
g(λ |T,r,A,B) =
The posterior mean and posterior variance are
E(Λ |T,r,A,B) =
Var(Λ |T,r,A,B) =
T r+1 λr exp(−λT)
. (14.17d)
γ(r + 1 |B T) − γ(r + 1 |AT)
γ(r + 2 |B T) − γ(r + 2 |AT)
T � � , (14.17e)
γ(r + 1 |B T) − γ(r + 1 |AT)
γ(r + 3 |B T) − γ(r + 3 |AT)
T 2 � γ(r + 1 |B T) − γ(r + 1 |AT) �
− � E(Λ |T,r,A,B) � 2 . (14.17f)
(14.17e) is the BAYESIAN estimator with respect to squared–error loss. The estimator
�λ = r/T is the mode of (14.17d).
The limits λℓ and λu of an equal–tail 100(1 − α)% credible interval
are found from solving
and
Pr(Λ < λℓ |T,r,A,B) =
Pr(Λ > λu |T,r,A,B) =
λℓ ≤ Λ ≤ λu
�λℓ
A
T r+1 λ r exp(−λT)
γ(r + 1 |B T) − γ(r + 1 |AT) dλ
= γ(r + 1 |λℓ T) − γ(r + 1 |AT)
γ(r + 1 |B T) − γ(r + 1 |AT)
�B
λu
= α
2
T r+1 λ r exp(−λT)
γ(r + 1 |B T) − γ(r + 1 |AT) dλ
= γ(r + 1 |B T) − γ(r + 1 |λu T)
γ(r + 1 |B T) − γ(r + 1 |AT)
using an incomplete gamma subroutine and a search program.
= α
2
(14.18a)
(14.18b)