Benktander type I distribution

Short description: Probability distribution in actuarial science

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	$\frac{-\sqrt{b}+a{e}^{\frac{(a-1{)}^2}{4b}}\sqrt{\pi }\text{<mrow data-mjx-texclass="ORD">erfc</mrow>}\left(\frac{a-1}{2\sqrt{b}}\right)}{a^2\sqrt{b}}$

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). The distribution of the first type is "close" to the log-normal distribution (Kleiber Kotz).

See also

• log-normal distribution
• Benktander type II distribution

References

• Kleiber, Christian; Kotz, Samuel (2003). "7.4 Benktander Distributions". Statistical Size Distributions in Economics and Actuarial Science. Wiley Series and Probability and Statistics. John Wiley & Sons. pp. 247–250. ISBN 9780471457169. 
• Benktander, Gunnar; Segerdahl, Carl-Otto (1960). "On the Analytical Representation of Claim Distributions with Special Reference to Excess of Loss Reinsurance". Proceedings of the XVIth International Congress of Actuaries, Brussels, 1960: 626–646. 
• Benktander, Gunnar (1970). "Schadenverteilungen nach Grösse in der Nicht-Lebensversicherung" (in German). Bulletin of the Swiss Association of Actuaries: 263–283.