双伽玛分布#
双伽玛是伽玛分布的有符号版本。对于 \(\alpha>0:\)
\begin{eqnarray*} f\left(x;\alpha\right) & = & \frac{1}{2\Gamma\left(\alpha\right)}\left|x\right|^{\alpha-1}e^{-\left|x\right|}\\ F\left(x;\alpha\right) & = & \left\{ \begin{array}{ccc} \frac{1}{2}-\frac{\gamma\left(\alpha,\left|x\right|\right)}{2\Gamma\left(\alpha\right)} & & x\leq0\\ \frac{1}{2}+\frac{\gamma\left(\alpha,\left|x\right|\right)}{2\Gamma\left(\alpha\right)} & & x>0 \end{array} \right.\\ G\left(q;\alpha\right) & = & \left\{ \begin{array}{ccc} -\gamma^{-1}\left(\alpha,\left|2q-1\right|\Gamma\left(\alpha\right)\right) & & q\leq\frac{1}{2}\\ \gamma^{-1}\left(\alpha,\left|2q-1\right|\Gamma\left(\alpha\right)\right) & & q>\frac{1}{2} \end{array} \right.\end{eqnarray*}
\[M\left(t\right)=\frac{1}{2\left(1-t\right)^{a}}+\frac{1}{2\left(1+t\right)^{a}}\]
\begin{eqnarray*} \mu=m_{n} & = & 0\\ \mu_{2} & = & \alpha\left(\alpha+1\right)\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & \frac{\left(\alpha+2\right)\left(\alpha+3\right)}{\alpha\left(\alpha+1\right)}-3\\ m_{d} & = & \mathrm{NA}\end{eqnarray*}