3.14 The Cornish-Fisher Expansion
The Cornish-Fisher expansion is a formula for approximating quantiles of a random variable based only on its first few cumulants. In this section, we define cumulants, specify the Cornish-Fisher expansion, and present an example.
The cumulants of a random variable X are conceptually similar to its moments. They are defined, somewhat abstrusely, as those values κr such that the identity
holds for all t. Cumulants of a random variable X can—see Stuart and Ord (1994)—be expressed in terms of its mean μ and central moments μr = E[(X – μ)r ]. Expressions for the first five cumulants are
3.14.2 Cornish-Fisher Expansion With
Suppose X has mean 0 and standard deviation 1. Cornish and Fisher (1937) provide an expansion for approximating the q-quantile, , of X based upon its cumulants. Using the first five cumulants, the expansion is
Although [3.206] applies only if X has mean 0 and standard deviation 1, we can still use it to approximate quantiles if X has some other mean μ and standard deviation σ. Simply define the normalization of X as
which has mean 0 and standard deviation 1. Central moments of X* can be calculated from central moments of X with
where σ = is the standard deviation of X. Apply the Cornish-Fisher expansion to obtain the q-quantile x* of X*. The corresponding q-quantile x of X is then
The Cornish-Fisher expansion [3.206] yields the .10-quantile of Y* as –1.123. Applying [3.209], we obtain the .10-quantile of Y as –5.029.
Using a spreadsheet and inputs from Exhibit 3.27, reproduce the results from the example of this section.