# 埃奇沃斯級數

## Gram-Charlier A series

Gram-Charlier A series 的主要想法，是把待逼近分布（以F 為它的密度函數）的特徵方程，寫成另一個已知分布的特徵方程的展式，再經過傅立葉變換的逆變換，就可以求得F 的展式。

${\displaystyle f(t)=\exp \left[\sum _{r=1}^{\infty }(\kappa _{r}-\gamma _{r}){\frac {(it)^{r}}{r!}}\right]\psi (t)\,.}$

${\displaystyle F(x)=\exp \left[\sum _{r=1}^{\infty }(\kappa _{r}-\gamma _{r}){\frac {(-D)^{r}}{r!}}\right]\Psi (x)\,.}$

${\displaystyle F(x)=\exp \left[\sum _{r=3}^{\infty }\kappa _{r}{\frac {(-D)^{r}}{r!}}\right]{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left[-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right]\,.}$

${\displaystyle F(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left[-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right]\left[1+{\frac {\kappa _{3}}{3!\sigma ^{3}}}H_{3}\left({\frac {x-\mu }{\sigma }}\right)+{\frac {\kappa _{4}}{4!\sigma ^{4}}}H_{4}\left({\frac {x-\mu }{\sigma }}\right)\right]\,,}$

## 延伸阅读

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