Parameters a
>
0
{\displaystyle a>0}
(real)
0
<
b
≤
1
{\displaystyle 0 Support x
≥
1
{\displaystyle x\geq 1} PDF e
a
b
(
1
−
x
b
)
x
b
−
2
(
a
x
b
−
b
+
1
)
{\displaystyle e^{{\frac {a}{b}}(1-x^{b})}x^{b-2}\left(ax^{b}-b+1\right)} CDF 1
−
x
b
−
1
e
a
b
(
1
−
x
b
)
{\displaystyle 1-x^{b-1}e^{{\frac {a}{b}}(1-x^{b})}} Mean 1
+
1
a
{\displaystyle 1+{\frac {1}{a}}} Median {
log
(
2
)
a
+
1
if
b
=
1
(
(
1
−
b
a
)
W
(
2
b
1
−
b
a
e
a
1
−
b
1
−
b
)
)
1
b
otherwise
{\displaystyle {\begin{cases}{\frac {\log(2)}{a}}+1&{\text{if}}\ b=1\\\left(\left({\frac {1-b}{a}}\right)\mathbf {W} \left({\frac {2^{\frac {b}{1-b}}ae^{\frac {a}{1-b}}}{1-b}}\right)\right)^{\tfrac {1}{b}}&{\text{otherwise}}\ \end{cases}}}
Where
W
(
x
)
{\displaystyle \mathbf {W} (x)}
is the Lambert W function | ||
The Benktander type II distribution, also called the Benktander distribution of the second kind, is one of two distributions introduced by Gunnar Benktander (1970) to model heavy-tailed losses commonly found in non-life/casualty actuarial science, using various forms of mean excess functions (Benktander & Segerdahl 1960). This distribution is "close" to the Weibull distribution (Kleiber & Kotz 2003).
References
Benktander type II distribution Wikipedia(Text) CC BY-SA