Cornish–Fisher expansion

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

Herein, the factor is only dependent on the considered quantile and corresponds to the value of the inverted distribution function of the standard normal distribution to the site.

The Cornish -Fisher expansion now takes into account the skewness and curvature of a distribution, which, of course, other quantiles than for the normal distribution yield, skewness and kurtosis which each is 0. Here, the factor is adjusted by means of

( Cornish - Fisher - assessment).

The calculation of the Quantilsfunktion is thus

The method allows, among other things, a better estimate of quantilsbezogenen risk measures, such as the Value at Risk when the normal distribution assumption is violated.

• Random variable