Benktander type II distribution

Benktander type II distribution

Parameters	$a>0$ (real) $0<b\leq1$ (real)
Support	$x \geq 1$
PDF	$e^{\frac{a}{b}(1 - x^b)}x^{b-2}\left(ax^b - b + 1\right)$
CDF	$1 - x^{b-1}e^{\frac{a}{b}(1 - x^b)}$
Mean	$1+\frac{1}{a}$
Median	$\begin{cases} \frac{\log(2)}{a}+1 & \text{if}\ b=1 \\ \left( \left(\frac{1-b}{a}\right)\mathbf{W}\left(\frac{ 2^{\frac{b}{1-b}} a e^{\frac{a}{1-b}} }{1-b} \right) \right)^{\tfrac{1}{b}} & \text{otherwise}\ \end{cases}$ Where $\mathbf{W}(x)$ is the Lambert W function[note 1]
Mode	$1$
Variance	$\frac{-b + 2ae^{\frac{a}{b}}\mathbf{E}_{1-\frac{1}{b}}\left(\frac{a}{b}\right)}{a^2 b}$ Where $\mathbf{E}_n(x)$ is the generalized Exponential integral[note 1]

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).

Notes

1. From Wolfram Alpha

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" [Loss Distributions by Size in Non-life Insurance]. Bulletin of the Swiss Association of Actuaries (in German): 263–283.