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In [[statistics]], the '''Champernowne distribution''' is a symmetric, [[continuous probability distribution]], describing [[random variable]]s that take both positive and negative values. It is a generalization of the [[logistic distribution]] that was introduced by [[D. G. Champernowne]].<ref name=KK>{{cite book|author=C. Kleiber and S. Kotz|title=Statistical Size Distributions in Economics and Actuarial Sciences|publisher=Wiley| location=New York|year = 2003|isbn=}} [http://books.google.com/books?id=7wLGjyB128IC&lpg=PA241&dq=Champernowne%20distribution&pg=PA240#v=onepage&q=Champernowne%20distribution&f=false   Section 7.3 "Champernowne Distribution."]</ref><ref name=Champ52>{{cite journal|author=Champernowne, D. G.| journal=Econometrica | title= The graduation of income distributions| year=1952 | volume=20 | page = 591–614 |jstor=1907644}}</ref><ref name=Champ53>{{cite journal|title=A Model of Income Distribution
|first1 = D. G. |last1=Champernowne|journal = The Economic Journal |volume=63 |issue=250 | year=1953 | page= 318–351 |jstor= 2227127 }}</ref> Champernowne developed the distribution to describe the logarithm of income.<ref name=Champ52/>

==Definition==
The Champernowne distribution has a [[probability density function]] given by

:<math>
 f(y;\alpha, \lambda, y_0 ) = \frac{n}{\cosh[\alpha(y - y_0)] + \lambda}, \qquad -\infty < y < \infty,
</math>

where <math> \alpha, \lambda, y_0</math> are positive parameters, and ''n'' is the normalizing constant, which depends on the parameters. The density may be rewritten as
:<math>
 f(y) = \frac{n}{1/2 e^{\alpha(y-y_0)} + \lambda + 1/2 e^{-\alpha(y-y_0)}},
</math>

using the fact that <math> \cosh y = (e^y + e^{-y})/2.</math> 

===Properties===

The density ''f''(''y'') defines a symmetric distribution with median ''y''<sub>0</sub>, which has tails somewhat heavier than a normal distribution.

===Special cases===
In the special case <math>\lambda=1</math> it is the [[Burr distribution|Burr Type XII]] density.

When <math> y_0 = 0, \alpha=1, \lambda=1 </math>, 
:<math>
 f(y) = \frac{1}{e^y + 2 + e^{-y}} = \frac{e^y}{(1+e^y)^2},
</math>

which is the density of the standard [[logistic distribution]]. 

== Distribution of income ==

If the distribution of ''Y'', the logarithm of income, has a Champernowne distribution, then the density function of the income ''X''&nbsp;=&nbsp;exp(''Y'') is<ref name=KK/> 
:<math>
 f(x) = \frac{n}{x [1/2(x/x_0)^{-\alpha} + \lambda + a/2(x/x_0)^\alpha ]}, \qquad x > 0,
</math>

where ''x''<sub>0</sub> = exp(''y''<sub>0</sub>) is the median income. If λ = 1, this distribution is often called the [[Fisk distribution]],<ref>Fisk, P. R. (1961). "The graduation of income distributions". ''Econometrica'', 29, 171–185.</ref> which has density
:<math>
 f(x) = \frac{\alpha x^{\alpha - 1}}{x_0^\alpha [1 + (x/x_0)^\alpha]^2}, \qquad x > 0.
</math>

==See also==
*[[Generalized logistic distribution]]

== References ==
{{Reflist}}


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{{ProbDistributions|continuous-infinite}}

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[[Category:Continuous distributions]]
[[Category:Probability distributions]]