View source for Cornish–Fisher expansion

{{short description|Infinite series used to approximate quantiles of probability distributions}}

In [[probability theory]], the '''Cornish–Fisher expansion''' is an [[asymptotic expansion]] used to approximate the [[quantile]]s of a [[probability distribution]] based on its [[cumulant]]s.<ref name=Cornish1937>{{cite journal
| last       = Cornish
| first      = E. A.
| last2      = Fisher
| first2     = Ronald A.
| authorlink2= Ronald Fisher
| year       = 1938
| title      = Moments and Cumulants in the Specification of Distributions
| journal    = Revue de l'Institut International de Statistique / Review of the International Statistical Institute
| volume     = 5
| issue      = 4
| pages      = 307–320
| doi        = 10.2307/1400905
| jstor        = 1400905
| url= https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15229/1/148.pdf
| hdl= 2440/15229
| hdl-access= free
}}
</ref><ref name=Fisher1960>{{cite journal
| last       = Fisher
| first      = Ronald A.
| authorlink = Ronald Fisher
| last2      = Cornish
| first2     = E. A.
| year       = 1960
| title      = The Percentile Points of Distributions Having Known Cumulants
| journal    = [[Technometrics]]
| volume     = 2
| issue      = 2
| pages      = 209–225
| doi        = 10.2307/1266546
| jstor        = 1266546
| url= https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15277/1/281.pdf
| hdl= 2440/15277
| hdl-access= free
}}
</ref><ref name=Abramowitz935>{{cite book
| last1                 = Abramowitz
| first1                = Milton
| author-link1          = Milton Abramowitz
| last2                 = Stegun
| first2                = Irene
| author-link2          = Irene Stegun
| year                  = 1964
| chapter               = 26. Probability Functions
| title                 = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
| publisher             = [[Dover Publications]]
| page                  = 935
| chapter-url           = http://people.math.sfu.ca/~cbm/aands/page_935.htm
| accessdate = September 17, 2014
| title-link                = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
}}
</ref><ref name=Inefficiency2017>{{cite journal
| last1                 = Martin
| first1                = Douglas
| last2                 = Arora
| first2                = Rohit
| year                  = 2017
| title                 = Inefficiency and bias of modified value-at-risk and expected shortfall
| journal    = [[Journal of Risk]]
| volume     = 19
| issue      = 6
| pages      = 59–84
| doi        = 10.21314/JOR.2017.365
}}
</ref>

It is named after [[Edmund Alfred Cornish|E. A. Cornish]] and [[Ronald Fisher|R. A. Fisher]], who first described the technique in 1937.<ref name=Cornish1937 />

The analogous expansion using [[L-moment]]s is due to Sillitto.

==Definition==

For a random variable '''''X''''' with mean μ, variance σ², and cumulants κ<sub>''n''</sub>, its quantile ''y<sub>p</sub>'' at order-of-quantile ''p'' can be estimated as <math>y_p \approx \mu + \sigma w_p</math> where:<ref name=Abramowitz935 />

:<math>
\begin{align}
w_p &=& x &+ \left[\gamma_1 h_1(x)\right]\\
&&&+ \left[\gamma_2 h_2(x) + \gamma_1^2 h_{11}(x)\right]\\
&&&+ \left[\gamma_3 h_3(x) + \gamma_1\gamma_2 h_{12}(x) + \gamma_1^3 h_{111}(x)\right]\\
&&&+ \cdots\\
\end{align}
</math>

:<math>
\begin{align}
x &= \Phi^{-1}(p)\\
\gamma_{r - 2} &= \frac{\kappa_r}{\kappa_2^{r/2}};\; r \in \{3, 4, \ldots\}\\
h_1(x) &= \frac{\mathrm{He}_2(x)}{6}\\
h_2(x) &= \frac{\mathrm{He}_3(x)}{24}\\
h_{11}(x) &= -\frac{\left[2\mathrm{He}_3(x) + \mathrm{He}_1(x)\right]}{36}\\
h_3(x) &= \frac{\mathrm{He}_4(x)}{120}\\
h_{12}(x) &= -\frac{\left[\mathrm{He}_4(x) + \mathrm{He}_2(x)\right]}{24}\\
h_{111}(x) &= \frac{\left[12\mathrm{He}_4(x) + 19\mathrm{He}_2(x)\right]}{324}
\end{align}
</math>

where He<sub>''n''</sub> is the ''n''<sup>th</sup> probabilists' [[Hermite polynomial]]. The values ''γ''<sub>1</sub> and ''γ''<sub>2</sub> are the random variable's [[skewness]] and (excess) [[kurtosis]] respectively. The value(s) in each set of brackets are the terms for that level of polynomial estimation, and all must be calculated and combined for the Cornish–Fisher expansion at that level to be valid.

==Example==
Let '''''X''''' be a random variable with mean 10, variance 25, skew 5, and excess kurtosis of 2. We can use the first two bracketed terms above, which depend only on skew and kurtosis, to estimate quantiles of this random variable. For the 95th percentile, the value for which the standard normal cumulative distribution function is 0.95 is 1.644854, which will be&nbsp;''x''. The ''w'' weight can be calculated as:

:<math>
\begin{align}
1.644854 &+ 5\cdot\frac{1.644854^2 - 1}{6}\\
&+ 2\cdot\frac{1.644854^3 - 3\cdot 1.644854}{24} - 5^2\frac{2\cdot 1.644854^3 - 5\cdot 1.644854}{36}
\end{align}
</math>

or about 2.55621. So the estimated 95th percentile of '''''X''''' is 10&nbsp;+&nbsp;5×2.55621 or about 22.781. For comparison, the 95th percentile of a normal random variable with mean 10 and variance 25 would be about 18.224; it makes sense that the normal random variable has a lower 95th percentile value, as the normal distribution has no skew or excess kurtosis, and so has a thinner tail than the random variable&nbsp;'''''X'''''.

==References==
{{reflist}}

{{DEFAULTSORT:Cornish-Fisher expansion}}
[[Category:Logical expressions]]
[[Category:Statistical deviation and dispersion]]
[[Category:Statistical approximations]]
[[Category:Asymptotic theory (statistics)]]