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