Benktander type I distribution

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Benktander distribution of the first kind
Parameters a>0 (real)
b>0 (real
Support x\geq 1
PDF  \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)}
CDF  1 - \left(1+\frac{2b}{a}\log x\right)x^{-\left(a + 1 + b\log x\right)}
Mean 1+\tfrac{1}{a}
Variance  \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}}[note 1]

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 lognormal distribution (Kleiber & Kotz 2003).



  • 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.