Log Double Exponential (Log-Laplace) Distribution#

One shape parameter $c>0$ . The support is $x\geq0$ .

\begin{array}{rcl} f (x; c) & = & {\begin{array}{ccc} \frac{c}{2} x^{c - 1} & 0 < x < 1 \\ \frac{c}{2} x^{- c - 1} & x \geq 1 \end{array} \\ F (x; c) & = & {\begin{array}{ccc} \frac{1}{2} x^{c} & 0 < x < 1 \\ 1 - \frac{1}{2} x^{- c} & x \geq 1 \end{array} \\ G (q; c) & = & {\begin{array}{ccc} {(2 q)}^{1 / c} & 0 \leq q < \frac{1}{2} \\ {(2 - 2 q)}^{- 1 / c} & \frac{1}{2} \leq q \leq 1 \end{array} \end{array}

$\begin{eqnarray*} f\left(x;c\right) & = & \left\{ \begin{array}{ccc} \frac{c}{2}x^{c-1} & & 0 < x < 1 \\ \frac{c}{2}x^{-c-1} & & x \geq 1 \end{array} \right. \\ F\left(x;c\right) & = & \left\{ \begin{array}{ccc} \frac{1}{2}x^{c} & & 0 < x < 1 \\ 1-\frac{1}{2}x^{-c} & & x \geq 1 \end{array} \right. \\ G\left(q;c\right) & = & \left\{ \begin{array}{ccc} \left(2q\right)^{1/c} & & 0 \leq q < \frac{1}{2} \\ \left(2-2q\right)^{-1/c} & & \frac{1}{2} \leq q \leq 1 \end{array} \right. \end{eqnarray*}$

h [X] = \log (\frac{2 e}{c})

$h\left[X\right]=\log\left(\frac{2e}{c}\right)$

Implementation: scipy.stats.loglaplace