Benktander type II distribution

name =Benktander type II distribution| type =density| pdf_image = [[Image:Benktander2PDF.svg|325px]]| cdf_image = [[Image:Benktander2CDF.svg|325px]]| parameters =a0 (real) 0 (real) | support =x \geq 1| pdf = e^{\frac{a}{b}(1 - x^b)}x^{b-2}\left(ax^b - b + 1\right) | cdf = 1 - x^{b-1}e^{\frac{a}{b}(1 - x^b)} | mean =1+\frac{1}{a}| median =\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} Where \mathbf{W}(x) is the Lambert W functionFrom Wolfram Alpha | mode = 1 | variance = \frac{-b + 2ae^{\frac{a}{b}}\mathbf{E}_{1-\frac{1}{b}}\left(\frac{a}{b}\right)}{a^2 b} Where \mathbf{E}_n(x) is the generalized Exponential integral | skewness =| kurtosis =| entropy =| mgf =| char =| The Benktander type II distribution, also called the Benktander distribution of the second kind, is one of two distributions introduced by Gunnar to model heavy-tailed losses commonly found in non-life/casualty actuarial science, using various forms of mean excess functions . This distribution is "close" to the Weibull distribution .