Benktander type I distribution

Benktander distribution of the first kind

Parameters	${\displaystyle a>0}$ (real) ${\displaystyle 0<b\leq {\frac {a(a+1)}{2}}}$ (real)
Support	${\displaystyle x\geq 1}$
PDF	${\displaystyle \left(\left[\left(1+{\frac {2b\log x}{a}}\right)\left(1+a+2b\log x\right)\right]-{\frac {2b}{a}}\right)x^{-\left(2+a+b\log x\right)}}$
CDF	${\displaystyle 1-\left(1+{\frac {2b\log x}{a}}\right)x^{-\left(a+1+b\log x\right)}}$
Mean	${\displaystyle 1+{\tfrac {1}{a}}}$
Variance	${\displaystyle {\frac {-{\sqrt {b}}+ae^{\frac {(a-1)^{2}}{4b}}{\sqrt {\pi }}\;{\textrm {erfc}}\left({\frac {a-1}{2{\sqrt {b}}}}\right)}{a^{2}{\sqrt {b}}}}}$[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 log-normal distribution (Kleiber & Kotz 2003).

• log-normal distribution
• Benktander type II distribution

• 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.