en>David Eppstein: /* References */ split off the two different kinds of references

2013-09-17T07:04:32Z

References: split off the two different kinds of references

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The '''Cornish–Fisher expansion''' is a mathematical expression used to approximate the [[quantile]]s of a [[random variable]] based only on its first few [[cumulant]]s.<ref name=Cornish1937>Cornish EA and Fisher RA (1938) Moments and cumulants in the specification of distributions. Revue de l’Institut Internat. de Statistique. 5: 307–322</ref><ref name=Fisher1960>Fisher RA and Cornish EA (1960) The percentile points of distributions having known cumulants. Technometrics 2: 209–225</ref><ref name=Abramowitz1985>Abramowitz M and Stegun I (1965) Handbook of mathematical functions, with formulas, graphs and mathematical tables. Dover Publications, New York</ref>

==Definition==

Let ''x'' be a random variable with a density function ''f''(''x'') with a mean of zero and a variance of 1. Let ''β''1 be the [[skewness]] of this distribution and let ''β''2 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

: <math> \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 ) </math>

where ''ω''''α'' is the corresponding value for ''f''(''x'').

==References==
{{reflist}}

{{DEFAULTSORT:Cornish-Fisher expansion}}
[[Category:Logical expressions]]
[[Category:Statistical deviation and dispersion]]

Surjunctive group - Revision history

en>David Eppstein: /* References */ split off the two different kinds of references