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

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

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 it is the Burr Type XII density.

When ,

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]

where x0 = exp(y0) is the median income. If λ = 1, this distribution is often called the Fisk distribution,[4] which has density

See also

References

  1. 1 2 C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."
  2. 1 2 Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644.
  3. Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal. 63 (250): 318–351. doi:10.2307/2227127. JSTOR 2227127.
  4. Fisk, P. R. (1961). "The graduation of income distributions". Econometrica. 29 (2): 171–185. doi:10.2307/1909287. JSTOR 1909287.
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