功率正态分布¶
只有一个形状参数的正态分布的推广 \(c>0\) 和支持 \(x\geq0\) 。
\Begin{eqnarray*}f\Left(x；c\Right)&=&c\phi\left(x\right)\left(\Phi\left(-x\right)\right)^{c-1}\\
F\Left(x；c\Right)&=&1-\Left(\Phi\Left(-x\Right)\Right)^{c}\\
g\Left(q；c\Right)&=&-\Phi^{-1}\left(\left(1-q\right)^{1/c}\right)\end{eqnarray*}
\[\mu_{n}^{\prime}=\left(-1\right)^{n}\int_{0}^{1}\left [\Phi^{{-1}}\left(y^{{1/c}}\right)\right] ^{n}天\]
\BEGIN{eqnarray *}} \mu & = & \mu_{{1}}^{{\prime}}\\ \mu_{{2}} & = & \mu_{{2}}^{{\prime}}-\mu^{{2}}\\ \gamma_{{1}} & = & \frac{{\mu_{{3}}^{{\prime}}-3\mu\mu_{{2}}-\mu^{{3}}}}{{\mu_{{2}}^{{3/2}}}}\\ \gamma_{{2}} & = & \frac{{\mu_{{4}}^{{\prime}}-4\mu\mu_{{3}}-6\mu^{{2}}\mu_{{2}}-\mu^{{4}}}}{{\mu_{{2}}^{{2}}}}-3\end{{eqnarray* }
为 \(c=1\) 这将减少到正态分布。