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. Section 7.3 "Champernowne Distribution." 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(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 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(Y) is
where x0 = exp(y0) is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density
See also Generalized logistic distribution
References