# Benktander Weibull distribution

Parameters Probability density function File:No image available Cumulative distribution function File:No image available $a>0$ (real) $0 (real) x ∈ [1, ∞) $e^{\tfrac{a(1-x^b)}{b}} x^{-2+b} (1-b+a x^b)$ $1 - e^{\tfrac{a(1-x^b)}{b}} x^{-1+b}$ $1+\tfrac{1}{a}$ $\begin{cases} 1+\frac{Log[2]}{a} & \text{if}\ b=1 \\ \left( \frac{1-b}{a} W\left(\frac{ 2^{\frac{b}{1-b}} a e^{\frac{a}{1-b}} }{1-b} \right) \right)^{\tfrac{1}{b}} & \text{otherwise}\ \end{cases}$ Where $W(x)$ is the Lambert W function $1$ $\frac{-1+\frac{2a e^{\tfrac{a}{b}} {\rm E}({1-\tfrac{1}{b}},\tfrac{a}{b}) }{b}}{a^2}$ Where ${\rm E}(n,x)$ is shorthand for ${\rm E}_n(x)$ the generalized Exponential integral

The Benktander-Weibull distribution, also known as the Benktander distribution of the second kind has the property that if

$X\ \sim\ BW2(a,b)$

then the mean residual life function equals

$\mathrm{E} \left (X\ - s | X\ > s \right ) = \frac{s^{1-b}}{a}, \qquad \forall s \ge 1.$
• BW2(a,1) ~ 1+ Exp(a)

## References

• 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.
• Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997). Modelling Extremal Events: For Insurance and Finance. Stochastic Modelling and Applied Probability 33. Springer. p. 35. ISBN 9783540609315.