The truncated Cornish–Fisher inverse expansion is well known and has been used to approximate value-at-risk and conditional value-at-risk. The following are also known. The expansion is available only for a limited range of skewness and kurtosis. The distribution approximation it gives is poor for larger values of skewness or kurtosis. We develop a computational method to find a unique corrected Cornish–Fisher distribution efficiently for a wide range of skewness and kurtosis. We show it has a unimodal density and a quantile function that is twice continuously differentiable as a function of mean, variance, skewness and kurtosis. We extend the univariate distribution to a multivariate Cornish–Fisher distribution and show it can be used together with estimation-error reduction methods to improve risk estimation. We show how to test goodness-of-fit. We apply the Cornish–Fisher distribution to fit hedge-fund returns and estimate conditional value-at-risk. We conclude that the Cornish–Fisher distribution is useful in estimating risk, especially in the multivariate case where we must deal with estimation error.

中文翻译：

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让 Cornish-Fisher 适合风险衡量

截断的 Cornish-Fisher 逆展开是众所周知的，并已被用于估算风险价值和条件风险价值。以下也是已知的。扩展仅适用于有限范围的偏度和峰度。对于较大的偏度或峰度值，它给出的分布近似很差。我们开发了一种计算方法，可以有效地为各种偏度和峰度找到唯一的校正康沃尔-费舍尔分布。我们证明它具有单峰密度和分位数函数，该函数作为均值、方差、偏度和峰度的函数可连续两次微分。我们将单变量分布扩展到多变量 Cornish-Fisher 分布，并表明它可以与减少估计误差的方法一起使用来改进风险估计。我们展示了如何测试拟合优度。我们应用 Cornish-Fisher 分布来拟合对冲基金的回报并估计有条件的风险价值。我们得出结论，Cornish-Fisher 分布在估计风险时很有用，尤其是在我们必须处理估计误差的多变量情况下。