Cornish-Fisher​ ​Expansion

It​ ​is​ ​​approximate​ ​method​ ​for​ ​deriving​ ​quantiles​ ​of​ ​any​ ​random​ ​variable​ ​distribution​ ​based​ ​on​ ​its cumulants.​ ​Cornish-Fisher​ ​Expansion​ ​can​ ​be​ ​used​ , for example, to​ ​calculate​ ​various​ ​Value​ ​At​ ​Risk​ ​(VaR)​ ​quantiles.

Cumulants​ ​κ​r​​ ​are​ ​an​ ​alternative​ ​expression​ ​of​ ​distribution​ ​moments​ ​µ​r​.​ ​For​ ​a​ ​cumulant​ ​κ​r of​ ​an order​ ​​r ​​it​ ​holds​ ​for​ ​all​ ​real​ ​​t ​(where​ ​µ​r​‘​ ​denotes​ ​raw​ ​moment):

Not​ ​trivial​ ​to​ ​derive​ ​at​ ​all,​ ​but​ ​can​ ​be​ ​generally​ ​expressed​ ​by​ ​recursion:

This​ ​leads​ ​to​ ​expression​ ​from​ ​raw​ ​moments​ ​(first​ ​4​ ​cumulants​ ​showed):

Alternatively​ ​derived​ ​from​ ​central​ ​moments​ ​(for​ ​r>1):

The​ ​Cornish-Fischer​ ​expansion​ ​for​ ​approximate​ ​determination​ ​of​ ​quantile​ ​x​q​​ ​builds​ ​on​ ​the​ ​variable​ ​X mean​ ​µ,​ ​standard​ ​deviation​ ​σ​ ​and​ ​cumulants​ ​κ​r​,​ ​with​ ​the​ ​help​ ​of​ ​quantiles​ ​of​ ​standard​ ​normal distribution​ ​N(0,1):

Moments

(Remember definition of probability density function f(x):

)

Moments​ ​are​ ​measure​ ​of​ ​​probability​ ​density​ (in general - measure of “shape of a set of points”). The​ ​n-th​ ​moment​ ​µ​n​ ​ ​of​ ​a​ ​real-valued​ ​continuous​ ​function​ ​f(x)​ ​about​ ​value​ ​c​ of a probability density function f(x) is defined:

The value c is typically set as the mean of a distribution, then the moments are called “central” moments. If c=0, the moment is called a “raw” moment and is marked as µn‘.

The​ ​zeroth​ ​(raw)​ ​moment​ ​is​ ​equal​ ​to​ ​1​ (total area under the probability density function f(x))

The​ ​first​ ​(raw)​ ​moment​ ​is​ ​the​ ​mean

The​ ​second​ ​(central)​ ​moment​ ​is​ ​the​ ​variance

For the higher moments, standardized variants are typically shown (divided by σ^n ) and are marked as µ ̃.

The​ ​third​ ​(standardized​ ​central)​ ​moment​ ​is​ ​skewness

The​ ​fourth​ ​(standardized​ ​central)​ ​moment​ ​is​ ​kurtosis