Log Double Exponential (Log-Laplace) Distribution#
One shape parameter \(c>0\). The support is \(x\geq0\).
 \begin{eqnarray*}
     f\left(x;c\right) & = & \left\{
                                 \begin{array}{ccc}
                                     \frac{c}{2}x^{c-1} &  & 0 < x < 1 \\
                                     \frac{c}{2}x^{-c-1} &  & x \geq 1
                                 \end{array}
                             \right. \\
     F\left(x;c\right) & = & \left\{
                                 \begin{array}{ccc}
                                     \frac{1}{2}x^{c} &  & 0 < x < 1 \\
                                     1-\frac{1}{2}x^{-c} &  & x \geq 1
                                 \end{array}
                             \right. \\
     G\left(q;c\right) & = & \left\{
                                 \begin{array}{ccc}
                                     \left(2q\right)^{1/c} &  & 0 \leq q < \frac{1}{2} \\
                                     \left(2-2q\right)^{-1/c} &  & \frac{1}{2} \leq q \leq 1
                                 \end{array}
                             \right.
 \end{eqnarray*}
\[h\left[X\right]=\log\left(\frac{2e}{c}\right)\]
Implementation: scipy.stats.loglaplace