 |
(1)
|
其中
 |  | ![x+[gamma_1h_1(x)]+[gamma_2h_2(x)+gamma_1^2h_(11)(x)]+[gamma_3h_3(x)+gamma_1gamma_2h_(12)(x)+gamma_1^3h_(111)(x)]+[gamma_4h_4(x)+gamma_2^2h_(22)(x)+gamma_1gamma_3h_(13)(x)+gamma_1^2gamma_2h_(112)(x)+gamma_1^4h_(1111)(x)]+...,](/images/equations/Cornish-FisherAsymptoticExpansion/Inline3.svg) |
(2)
|
其中
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  | ![-1/(36)[2He_3(x)+He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline12.svg) |
(5)
|
 |  |  |
(6)
|
 |  | ![-1/(24)[He_4(x)+He_2(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline18.svg) |
(7)
|
 |  | ![1/(324)[12He_4(x)+19He_2(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline21.svg) |
(8)
|
 |  |  |
(9)
|
 |  | ![-1/(384)[3He_5(x)+6He_3(x)+2He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline27.svg) |
(10)
|
 |  | ![-1/(180)[2He_5+3He_3(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline30.svg) |
(11)
|
 |  | ![1/(288)[14He_5(x)+37He_3(x)+8He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline33.svg) |
(12)
|
 |  | ![-1/(7776)[252He_5(x)+832He_3(x)+227He_1(x)]](/images/equations/Cornish-FisherAsymptoticExpansion/Inline36.svg) |
(13)
|
以及 埃爾米特多項式 的組合。
Cornish-Fisher 漸近展開
另請參閱
查理耶級數, 埃奇沃思級數使用 探索
參考文獻
Abramowitz, M. 和 Stegun, I. A. (編). 數學函式手冊，包含公式、圖表和數學表格，第 9 版。 紐約: Dover, p. 935, 1972.Cornish, E. A. 和 Fisher, R. A. "分佈規範中的矩和累積量。" 國際統計學會評論 4, 1-14, 1937. 重印於 Fisher, R. A. 數理統計貢獻。 紐約: Wiley, 1950.Wallace, D. L. "分佈的漸近近似。" 數學統計年鑑 29, 635-654, 1958.Wasow, W. "關於某些分佈到正態分佈的漸近變換。" 應用數學研討會論文集 VI，數值分析。 紐約: McGraw-Hill, pp. 251-259, 1956.在 中被引用
Cornish-Fisher 漸近展開引用為
Weisstein, Eric W. "Cornish-Fisher 漸近展開。" 來自 Web 資源。 https://mathworld.tw/Cornish-FisherAsymptoticExpansion.html