Benktander type I distribution

Benktander distribution of the first kind

Parameters	[math]\displaystyle{ a\gt 0 }[/math] (real) [math]\displaystyle{ 0 \lt b \leq \frac{a(a+1)}{2} }[/math] (real)
Support	[math]\displaystyle{ x\geq 1 }[/math]
PDF	[math]\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)} }[/math]
CDF	[math]\displaystyle{ 1 - \left(1+\frac{2b}{a}\log x\right)x^{-\left(a + 1 + b\log x\right)} }[/math]
Mean	[math]\displaystyle{ 1+\tfrac{1}{a} }[/math]
Variance	[math]\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}} }[/math]

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.