Benktander type II distribution

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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).


  1. ^ a b From Wolfram Alpha


  • 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.