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 quantiles of a probability distribution based on its cumulants.{{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
}}
| 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
}}
| 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
}}
| 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
}}
It is named after E. A. Cornish and R. A. Fisher, who first described the technique in 1937.
The analogous expansion using L-moments is due to Sillitto.
Definition
For a random variable X with mean μ, variance σ², and cumulants κn, its quantile yp at order-of-quantile p can be estimated as where:
:
\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}
:
\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}
where Hen is the nth probabilists' Hermite polynomial. The values γ1 and γ2 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 x. The w weight can be calculated as:
:
\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}
or about 2.55621. So the estimated 95th percentile of X is 10 + 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 X.
References
{{reflist}}
{{DEFAULTSORT:Cornish-Fisher expansion}}
Category:Statistical deviation and dispersion