Surjunctive group

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

${\omega }_{\alpha }=z_{\alpha }+\frac{1}{6}(z_{\alpha }^2-1){\beta }_1+\frac{1}{24}(z_{\alpha }^3-3z_{\alpha })({\beta }_2-3)-\frac{1}{36}(2z_{\alpha }^3-5z_{\alpha }){\beta }_1^2-\frac{1}{24}(z_{\alpha }^4-5z_{\alpha }^2+2){\beta }_1({\beta }_2-3)$

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

References

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