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. Champernowne developed the distribution to describe the logarithm of income.

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 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 hyperbolic secant distribution.
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 is
where x0 = exp is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density