Support x ≥ 1 {\displaystyle x\geq 1} PDF ( [ ( 1 + 2 b log x a ) ( 1 + a + 2 b log x ) ] − 2 b a ) x − ( 2 + a + b log x ) {\displaystyle \left(\left[\left(1+{\frac {2b\log x}{a}}\right)\left(1+a+2b\log x\right)\right]-{\frac {2b}{a}}\right)x^{-\left(2+a+b\log x\right)}} CDF 1 − ( 1 + 2 b a log x ) x − ( a + 1 + b log x ) {\displaystyle 1-\left(1+{\frac {2b}{a}}\log x\right)x^{-\left(a+1+b\log x\right)}} Mean 1 + 1 a {\displaystyle 1+{\tfrac {1}{a}}} Variance − b + a e ( a − 1 ) 2 4 b π erfc ( a − 1 2 b ) a 2 b {\displaystyle {\frac {-{\sqrt {b}}+ae^{\frac {(a-1)^{2}}{4b}}{\sqrt {\pi }}\;{\textrm {erfc}}\left({\frac {a-1}{2{\sqrt {b}}}}\right)}{a^{2}{\sqrt {b}}}}} | ||
The Benktander type I distribution 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). The distribution of the first type is "close" to the lognormal distribution (Kleiber & Kotz 2003).
References
Benktander type I distribution Wikipedia(Text) CC BY-SA