Gumbel distributions

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The Gumbel type I largest (resp. smallest) values distribution is usually referred as the extreme Value Type I maximum (resp. minimum) distribution. It is defined on the range [-\infty, \infty], using the following characteristics.

  • type I largest values
    \begin{matrix}


&f_x(x)  =  \alpha\exp\left[-\alpha(x-u)\right] \cdot\exp\left[-\exp\left[-\alpha(x-u)\right]\right] \\
 &F_x(x)  =  \exp\left[\exp\left[-\alpha(x-u)\right]\right] \\
 \end{matrix}

where \alpha and u are defined as:

\begin{matrix}

& \alpha  =  \frac{\pi}{\sigma\sqrt{6}} \qquad \alpha \ge 0 \\
 &u = \mu - \frac{1}{\sqrt{3}\alpha} 
 \end{matrix}
  • type I smallest values
    \begin{matrix}

&f_x(x)  =  \alpha\exp\left[\alpha(x-u)\right] \cdot\exp\left[-\exp\left[\alpha(x-u)\right]\right] \\
 &F_x(x)  =  1-\exp[-\left[\exp\left[\alpha(x-u)\right]\right]]  \\
 \end{matrix}

where \alpha and u are defined as:

\begin{matrix}

& \alpha  =  \frac{\pi}{\sigma\sqrt{6}} \qquad \alpha \ge 0 \\
 &u = \mu - \frac{1}{\sqrt{3}\alpha} 
 \end{matrix}