Benktander type I distribution

Benktander distribution of the first kind
Parameters	$a>0$ (real) $0<b\le \frac{a(a+1)}{2}$ (real)
Support	$x\ge 1$
PDF	$(\left[(1+\frac{2b\log x}{a})(1+a+2b\log x)\right]-\frac{2b}{a})x^{-(2+a+b\log x)}$
CDF	$1-(1+\frac{2b\log x}{a})x^{-(a+1+b\log x)}$
Mean	$1+\frac{1}{a}$
Variance	$\frac{-\sqrt{b}+a{e}^{\frac{(a-1{)}^2}{4b}}\sqrt{\pi }\text{erfc}\left(\frac{a-1}{2\sqrt{b}}\right)}{a^2\sqrt{b}}$

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 log-normal distribution (Kleiber & Kotz 2003).

See also

• log-normal distribution
• Benktander type II distribution