Champernowne distribution

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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[edit]

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[edit]

The density f(y) defines a symmetric distribution with median y0, which has tails somewhat heavier than a normal distribution.

Special cases[edit]

In the special case it is the Burr Type XII density.

When ,

which is the density of the standard logistic distribution.

Distribution of income[edit]

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[edit]

References[edit]

  1. ^ a b C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley.  Section 7.3 "Champernowne Distribution."
  2. ^ a b Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20: 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: 171–185. doi:10.2307/1909287.