|
|
| Line 1: |
Line 1: |
| 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
| | 49 yr old Translator Henricksen from Swan Lake, loves to spend some time pyrotechnics, [http://tinyurl.com/gtracing2hacks Gt Racing 2 Hack] and base jumping. Discovered some incredible places after working 6 days at Paris. |
| |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'' = 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}}
| |
| | |
| | |
| {{statistics-stub}}
| |
| {{ProbDistributions|continuous-infinite}}
| |
| | |
| {{DEFAULTSORT:Champernowne distribution}}
| |
| [[Category:Continuous distributions]]
| |
| [[Category:Probability distributions]]
| |
49 yr old Translator Henricksen from Swan Lake, loves to spend some time pyrotechnics, Gt Racing 2 Hack and base jumping. Discovered some incredible places after working 6 days at Paris.