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{{Distinguish|Davies distribution}} | |||
{{Probability distribution | | |||
name =Davis distribution| | |||
type =density| | |||
pdf_image =No image available| | |||
cdf_image =No image available| | |||
parameters =<math>b>0</math> [[scale parameter|scale]]<br/><math> n> 0</math> [[shape parameter|shape]]<br><math>\mu>0</math> [[location parameter|location]] | | |||
support =<math>x>\mu</math> | | |||
pdf =<math> \frac{ b^n {(x-\mu)}^{-1-n} }{ \left( e^{\frac{b}{x-\mu}} -1 \right) \Gamma(n) \zeta(n) } </math> <br> Where <math>\Gamma(n)</math> is the [[Gamma function]] and <math>\zeta(n)</math> is the [[Riemann zeta function]] | | |||
cdf =| | |||
mean =<math>\begin{cases} | |||
\mu + \frac{b\zeta(n-1)}{(n-1)\zeta(n)} & \text{if}\ n>2 \\ | |||
\text{Indeterminate} & \text{otherwise}\ \end{cases}</math> | | |||
median = | | |||
mode =| | |||
variance = <math>\begin{cases} | |||
\frac{ b^2 \left( -(n-2){\zeta(n-1)}^2+(n-1)\zeta(n-2)\zeta(n) \right)}{(n-2) {(n-1)}^2 {\zeta(n)}^2} & \text{if}\ n>3 \\ | |||
\text{Indeterminate} & \text{otherwise}\ \end{cases}</math> | | |||
skewness =| | |||
kurtosis =| | |||
entropy =| | |||
mgf =| | |||
char =| | |||
}} | |||
In [[statistics]], the '''Davis distributions''' are a family of [[continuous probability distribution]]s. It is named after Harold T. Davis (1892–1974), who in 1941 proposed this distribution to model income sizes. (''The Theory of Econometrics and Analysis of Economic Time Series''). It is a generalization of the [[Planck's law]] of radiation from [[statistical physics]]. | |||
==Definition== | |||
The [[probability density function]] of the Davis distribution is given by | |||
:<math>f(x;\mu,b,n)=\frac{ b^n {(x-\mu)}^{-1-n} }{ \left( e^{\frac{b}{x-\mu}} -1 \right) \Gamma(n) \zeta(n) } </math> | |||
where <math>\Gamma(n)</math> is the [[Gamma function]] and <math>\zeta(n)</math> is the [[Riemann zeta function]]. Here μ, ''b'', and ''n'' are parameters of the distribution, and ''n'' need not be an integer. | |||
==Background== | |||
In an attempt to derive an expression that would represent not merely the upper tail of the distribution of income, Davis required an appropriate model with the following properties<ref>[[#Kle03|Kleiber 2003]]</ref> | |||
* <math>f(\mu)=0 \, </math> for some <math>\mu>0 \,</math> | |||
* A modal income exists | |||
* For large ''x'', the density behaves like a [[Pareto distribution]]: | |||
::<math> f(x) \sim A {(x-\mu)}^{-\alpha-1} \, . </math> | |||
==Related distributions== | |||
* If <math>X \sim \mathrm{Davis}(b=1,n=4,\mu=0)\,</math> then<br><math>\tfrac{1}{X} \sim \mathrm{Planck} </math> ([[Planck's law]]) | |||
==Notes== | |||
{{reflist|2}} | |||
==References== | |||
* {{cite book |last=Kleiber |first=Christian |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=Wiley Series in Probability and Statistics |isbn=978-0-471-15064-0 |ref=Kle03}} | |||
*Davis, H. T. (1941). [http://cowles.econ.yale.edu/P/cm/m06/index.htm ''The Analysis of Economic Time Series'']. The Principia Press, Bloomington, Indiana [http://cowles.econ.yale.edu/P/cm/m06/m06-all.pdf Download book] | |||
*VICTORIA-FESER, Maria-Pia. (1993) [http://archive-ouverte.unige.ch/unige:6450 ''Robust methods for personal income distribution models'']. Thèse de doctorat : Univ. Genève, 1993, no. SES 384 (p. 178) | |||
{{ProbDistributions|continuous-semi-infinite}} | |||
[[Category:Continuous distributions]] | |||
[[Category:Probability distributions]] | |||
Revision as of 18:08, 19 November 2013
Template:Probability distribution
In statistics, the Davis distributions are a family of continuous probability distributions. It is named after Harold T. Davis (1892–1974), who in 1941 proposed this distribution to model income sizes. (The Theory of Econometrics and Analysis of Economic Time Series). It is a generalization of the Planck's law of radiation from statistical physics.
Definition
The probability density function of the Davis distribution is given by
where is the Gamma function and is the Riemann zeta function. Here μ, b, and n are parameters of the distribution, and n need not be an integer.
Background
In an attempt to derive an expression that would represent not merely the upper tail of the distribution of income, Davis required an appropriate model with the following properties[1]
- for some
- A modal income exists
- For large x, the density behaves like a Pareto distribution:
Related distributions
- If then
(Planck's law)
Notes
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References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Davis, H. T. (1941). The Analysis of Economic Time Series. The Principia Press, Bloomington, Indiana Download book
- VICTORIA-FESER, Maria-Pia. (1993) Robust methods for personal income distribution models. Thèse de doctorat : Univ. Genève, 1993, no. SES 384 (p. 178)
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