双伽马分布¶
双伽马是伽马分布的签名版本。为 \(\alpha>0:\)
\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{\γ\Left(\Alpha，\left|x\right|\right)}{2\Gamma\left(\alpha\right)}&&x\leq0\\
\FRAC{1}{2}+\FRAC{\γ\Left(\Alpha，\left|x\right|\right)}{2\Gamma\left(\alpha\right)}&&x>0
\end{数组}
\对。\\
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{数组}
\对。\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}&=&\mathm{na}\end{eqnarray*}