双重威布尔分布¶
这是威布尔分布的签名形式。有一个形状参数 \(c>0\) 。支持是 \(x\in\mathbb{{R}}\) 。
\Begin{eqnarray*}f\Left(x；c\Right)&=&\frac{c}{2}\left|x\right|^{c-1}\exp\left(-\left|x\right|^{c}\right)\\
f\Left(x；c\Right)&=&\Left\{
\BEGIN{array}{ccc}
\frac{1}{2}\exp\left(-\left|x\right|^{c}\right)&&x\leq0\\
1-\frac{1}{2}\exp\left(-\left|x\right|^{c}\right)&&x>0
\end{数组}
\对。\\
g\Left(q；c\Right)&=&\Left\{
\BEGIN{array}{ccc}
-\log^{1/c}\Left(\frac{1}{2q}\right)&&q\leq\frac{1}{2}\\
\log^{1/c}\Left(\frac{1}{2q-1}\right)&&q>\frac{1}{2}
\end{数组}
\对。\end{eqnarray*}
\[\begin{split}\MU_{n}^{\Prime}=\MU_{n}=\开始{案例}
\Gamma\Left(1+\frac{n}{c}\Right)&n\text{Even}\\
0&n\text{ODD}
\结束{案例}\end{split}\]
\BEGIN{eqnarray*}m_{n}=\m&=&0\\
\MU_{2}&=&\Gamma\Left(\frac{c+2}{c}\Right)\\
\Gamma_{1}&=&0\\
\Gamma_{2}&=&\frac{\Gamma\left(1+\frac{4}{c}\right)}{\Gamma^{2}\left(1+\frac{2}{c}\right)}\\
m_{d}&=&\text{na双峰}\end{eqnarray*}