Benktander type II distribution
From HandWiki
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Probability density function  | |||
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Cumulative distribution function  | |||
| Parameters |
[math]\displaystyle{ a\gt 0 }[/math] (real) [math]\displaystyle{ 0\lt b\leq1 }[/math] (real) | ||
|---|---|---|---|
| Support | [math]\displaystyle{ x \geq 1 }[/math] | ||
| [math]\displaystyle{ e^{\frac{a}{b}(1 - x^b)}x^{b-2}\left(ax^b - b + 1\right) }[/math] | |||
| CDF | [math]\displaystyle{ 1 - x^{b-1}e^{\frac{a}{b}(1 - x^b)} }[/math] | ||
| Mean | [math]\displaystyle{ 1+\frac{1}{a} }[/math] | ||
| Median |
[math]\displaystyle{ \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} }[/math] Where [math]\displaystyle{ \mathbf{W}(x) }[/math] is the Lambert W function[note 1] | ||
| Mode | [math]\displaystyle{ 1 }[/math] | ||
| Variance |
[math]\displaystyle{ \frac{-b + 2ae^{\frac{a}{b}}\mathbf{E}_{1-\frac{1}{b}}\left(\frac{a}{b}\right)}{a^2 b} }[/math] Where [math]\displaystyle{ \mathbf{E}_n(x) }[/math] 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). This distribution is "close" to the Weibull distribution (Kleiber Kotz).
See also
Notes
- ↑ 1.0 1.1 From Wolfram Alpha
References
- 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. https://archive.org/details/statisticalsized0000unse.
- 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" (in German). Bulletin of the Swiss Association of Actuaries: 263–283.
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