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. Champernowne developed the distribution to describe the logarithm of income.
The Champernowne distribution has a probability density function given by
where are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as
using the fact that
The density f(y) defines a symmetric distribution with median y0, which has tails somewhat heavier than a normal distribution.
In the special case it is the Burr Type XII density.
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
- C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."
- Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20: 591–614. doi:10.2307/1907644. JSTOR 1907644.
- Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal. 63 (250): 318–351. doi:10.2307/2227127. JSTOR 2227127.
- Fisk, P. R. (1961). "The graduation of income distributions". Econometrica. 29: 171–185. doi:10.2307/1909287.