The Weibull Distribution: A Handbook - Index of

36 2 Definition and properties of the WEIBULL distribution
2.2.3 Analysis of the reduced WEIBULL density 2
In this section the shape of the reduced density
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fU(u |c) = cu c−1 exp � − u c� ; c > 0, u ≥ 0 (2.19)
in its dependence on c will be investigated, especially the curvature (convex �= concave
upward) or concave (�= concave downward)), the mode and the points of inflection.
The behavior of fU(u |c) with u → 0 or u → ∞ is a follows:
lim
u→0 fU(u
⎧
⎪⎨
∞ for 0 < c < 1,
|c) =
⎪⎩
1
0
for
for
c = 1,
c > 1,
lim
u→∞ fU(u |c) = 0 ∀ c > 0.
The first and second derivatives of (2.19) with respect to u are
f ′ U (u |c) = cuc−2 � c − 1 − cu c� exp � − u c� , (2.20a)
f ′′
U (u |c) = cuc−3�2
+ c � c − 3 − 3(c − 1)u c 2
+ cu
c��
exp � − u c� . (2.20b)
The possible extreme values of fU(u |c) are given by the roots u ∗ of f ′ U
where the density is
(u |c) = 0:
u ∗ � �1/c c − 1
= , (2.21a)
c
fU(u ∗ � � (c−1)/c
c − 1
|c) = c . (2.21b)
ce
The possible points of inflection result from f ′′
U (u |c) = 0, a quadratic equation in v = cuc
with roots
u ∗∗ �
3(c − 1) ±
=
� �1/c (5c − 1)(c − 1)
, (2.22a)
2c
where the density is
fU(u ∗∗ �
3(c−1)±
|c)=c
� � (c−1)/c �
(5c−1)(c−1)
exp −
2c
3(c−1)±� �
(5c−1)(c−1)
.
2c
(2.22b)
When studying the behavior of fU(u |c), it is appropriate to make a distinction of six cases
regarding c.
2 Suggested readings for this section: FAREWELL/PRENTICE (1977), HAHN/GODFREY/RENZI (1960),
LEHMAN (1963), PLAIT (1962).