Benktander type II distribution

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).

Benktander type II distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle a>0}$ (real) ${\displaystyle 0<b\leq 1}$ (real)
Support	${\displaystyle x\geq 1}$
PDF	${\displaystyle e^{{\frac {a}{b}}(1-x^{b})}x^{b-2}\left(ax^{b}-b+1\right)}$
CDF	${\displaystyle 1-x^{b-1}e^{{\frac {a}{b}}(1-x^{b})}}$
Mean	${\displaystyle 1+{\frac {1}{a}}}$
Median	${\displaystyle {\begin{cases}{\frac {\log(2)}{a}}+1&{\text{if}}\ b=1\\\left(\left({\frac {1-b}{a}}\right)\mathbf {W} \left({\frac {2^{\frac {b}{1-b}}ae^{\frac {a}{1-b}}}{1-b}}\right)\right)^{\tfrac {1}{b}}&{\text{otherwise}}\ \end{cases}}}$ Where ${\displaystyle \mathbf {W} (x)}$ is the Lambert W function[note 1]
Mode	${\displaystyle 1}$
Variance	${\displaystyle {\frac {-b+2ae^{\frac {a}{b}}\mathbf {E} _{1-{\frac {1}{b}}}\left({\frac {a}{b}}\right)}{a^{2}b}}}$ Where ${\displaystyle \mathbf {E} _{n}(x)}$ is the generalized Exponential integral[note 1]