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# Benktander type II distribution

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

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).