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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 .
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
denotes the excess, i.e. H. the curvature going beyond the curvature of the normal distribution (overkurtosis).
(Cornish-Fisher estimate).
The calculation of the quantile function is thus
.
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.