# Benktander type II distribution

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