Cornish–Fisher expansion

The Cornish–Fisher expansion is a mathematical expression used to approximate the quantiles of a random variable based only on its first few cumulants.

Definition

Let x be a random variable with a density function f(x) with a mean of zero and a variance of 1. Let β₁ be the skewness of this distribution and let β₂ be its kurtosis. Let z be a normally distributed random variable and let z_α be the value of z at the α^th percentile.

As an illustration of this last definition when α = 0.95, z_α = 1.96

Then

$\omega_\alpha = z_\alpha + \frac{ 1 }{ 6 }( z_\alpha^2 - 1 ) \beta_1 + \frac{ 1 }{ 24 }(z_\alpha^3 - 3 z_\alpha )( \beta_2 - 3 ) - \frac{ 1 }{ 36 }( 2 z_\alpha^3 - 5 z_\alpha ) \beta_1^2 - \frac{ 1 }{ 24 } ( z_\alpha^4 - 5 z_\alpha^2 + 2 ) \beta_1 ( \beta_2 - 3 )$

where ω_α is the corresponding value for f(x).