# 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.[1][2][3]

## Definition

Let x be a random variable with a density function f(x) with a mean of zero and a variance of 1. Let z be a normally distributed random variable and let zα be the value of z at the αth percentile. Let $z_\alpha$ - $\alpha$-percent qunatile of the standard normal density function, β1 be the skewness of this distribution, and let β2 be its kurtosis. As an illustration of this last definition when α = 0.975, zα = 1.96

Then

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

where xα is the corresponding value for f(x), that is the value of x at its αth percentile.

## References

1. ^ Cornish EA and Fisher RA (1938) Moments and cumulants in the specification of distributions. Revue de l’Institut Internat. de Statistique. 5: 307–322
2. ^ Fisher RA and Cornish EA (1960) The percentile points of distributions having known cumulants. Technometrics 2: 209–225
3. ^ Abramowitz M and Stegun I (1965) Handbook of mathematical functions, with formulas, graphs and mathematical tables. Dover Publications, New York