Champernowne distribution
In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. G. Champernowne.[1][2][3] Champernowne developed the distribution to describe the logarithm of income.[2]
Definition
The Champernowne distribution has a probability density function given by
- [math]\displaystyle{ f(y;\alpha, \lambda, y_0 ) = \frac{n}{\cosh[\alpha(y - y_0)] + \lambda}, \qquad -\infty \lt y \lt \infty, }[/math]
where [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \cosh y = (e^y + e^{-y})/2. }[/math]
Properties
The density f(y) defines a symmetric distribution with median y0, which has tails somewhat heavier than a normal distribution.
Special cases
In the special case [math]\displaystyle{ \lambda=1 }[/math] it is the Burr Type XII density.
When [math]\displaystyle{ y_0 = 0, \alpha=1, \lambda=1 }[/math],
- [math]\displaystyle{ 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[1]
- [math]\displaystyle{ f(x) = \frac{n}{x [1/2(x/x_0)^{-\alpha} + \lambda + a/2(x/x_0)^\alpha ]}, \qquad x \gt 0, }[/math]
where x0 = exp(y0) is the median income. If λ = 1, this distribution is often called the Fisk distribution,[4] which has density
- [math]\displaystyle{ f(x) = \frac{\alpha x^{\alpha - 1}}{x_0^\alpha [1 + (x/x_0)^\alpha]^2}, \qquad x \gt 0. }[/math]
See also
References
- ↑ 1.0 1.1 C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."
- ↑ 2.0 2.1 Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica 20 (4): 591–614. doi:10.2307/1907644.
- ↑ Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal 63 (250): 318–351. doi:10.2307/2227127.
- ↑ Fisk, P. R. (1961). "The graduation of income distributions". Econometrica 29 (2): 171–185. doi:10.2307/1909287.
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