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. Section 7.3 "Champernowne Distribution."
/ref> Champernowne developed the distribution to describe the logarithm of income.

Definition

The Champernowne distribution has a probability density function given by

:$f(y;\alpha, \lambda, y_0 ) = \frac, \qquad -\infty < y < \infty,$

where $\alpha, \lambda, y_0$ are positive parameters, and ''n'' is the normalizing constant, which depends on the parameters. The density may be rewritten as
:$f(y) = \frac,$

using the fact that $\cosh x = \tfrac 1 2 (e^x + e^).$

Properties

The density ''f''(''y'') defines a symmetric distribution with median ''y''₀, which has tails somewhat heavier than a normal distribution.

Special cases

In the special case $\lambda = 0$ ($\alpha = \tfrac \pi 2, y_0 = 0$) it is the hyperbolic secant distribution.

In the special case $\lambda=1$ it is the Burr Type XII density.

When $y_0 = 0, \alpha=1, \lambda=1$,
:$f(y) = \frac = \frac,$

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
:$f(x) = \frac, \qquad x > 0,$

where ''x''₀ = exp(''y''₀) is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density
:$f(x) = \frac, \qquad x > 0.$

See also

* Generalized logistic distribution

References

{{DEFAULTSORT:Champernowne distribution
Continuous distributions