Benktander Weibull distribution

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Benktander-Weibull
Probability density function
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Cumulative distribution function
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Parameters a>0 (real)
0<b<=1 (real)
Support x ∈ [1, ∞)
pdf  e^{\tfrac{a(1-x^b)}{b}} x^{-2+b} (1-b+a x^b)
CDF  1 -  e^{\tfrac{a(1-x^b)}{b}} x^{-1+b}
Mean 1+\tfrac{1}{a}
Median \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
Mode  1
Variance  \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[edit]

  • 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.