Cornish Fisher method

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With the Cornish-Fisher method (according to EA Cornish and Ronald Aylmer Fisher ) the quantile of a distribution function can be estimated on the basis of the first four moments ( expected value , standard deviation , skewness and kurtosis ). The basis is the determination of a quantile of a normal distribution . In the case of a normal distribution with expected value , the quantiles of the distribution can be represented as ${\displaystyle E(X)}$

${\displaystyle Q_{\alpha }(X)=F_{x}^{-1}(\alpha )=E(X)+q_{\alpha }\cdot \sigma (X)}$.

The factor is only dependent on the quantile under consideration and corresponds to the value of the inverted distribution function of the standard normal distribution at the point . ${\displaystyle q_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

The Cornish-Fisher extension now takes into account the skewness and the curvature of a distribution, which of course results in different quantiles than in the normal distribution, whose skewness is 0 and kurtosis is 3. The factor is adjusted using ${\displaystyle \gamma }$ ${\displaystyle \delta }$ ${\displaystyle q_{\alpha }}$

${\displaystyle z_{\alpha }=q_{\alpha }+{\frac {1}{6}}(q_{\alpha }^{2}-1)\cdot \gamma +{\frac {1}{24}}(q_{\alpha }^{3}-3q_{\alpha })\cdot \varepsilon -{\frac {1}{36}}(2q_{\alpha }^{3}-5q_{\alpha })\cdot \gamma ^{2}}$

denotes the excess, i.e. H. the curvature going beyond the curvature of the normal distribution (overkurtosis). ${\displaystyle \varepsilon =\delta -3}$

(Cornish-Fisher estimate).

The calculation of the quantile function is thus

${\displaystyle Q_{\alpha }(X)=E(X)+z_{\alpha }\cdot \sigma (X)\,}$.

Among other things, the method enables a better estimation of quantile-related risk measures , e.g. B. the value at risk if the normal distribution hypothesis is violated.