对数双指数 (对数拉普拉斯) 分布#
一个形状参数 \(c>0\)。支持为 \(x\geq0\)。
\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*}
\[h\left[X\right]=\log\left(\frac{2e}{c}\right)\]