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

$f(y;\alpha ,\lambda ,y_{0})={\frac {n}{\cosh[\alpha (y-y_{0})]+\lambda }},\qquad -\infty <y<\infty ,$

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

$f(y)={\frac {n}{{\tfrac {1}{2}}e^{\alpha (y-y_{0})}+\lambda +{\tfrac {1}{2}}e^{-\alpha (y-y_{0})}}},$

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

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 $\lambda =0$ ($\alpha ={\tfrac {\pi }{2}},y_{0}=0$) it is the hyperbolic secant distribution.

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

When $y_{0}=0,\alpha =1,\lambda =1$,

$f(y)={\frac {1}{e^{y}+2+e^{-y}}}={\frac {e^{y}}{(1+e^{y})^{2}}},$

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

$f(x)={\frac {n}{x[1/2(x/x_{0})^{-\alpha }+\lambda +a/2(x/x_{0})^{\alpha }]}},\qquad x>0,$

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

$f(x)={\frac {\alpha x^{\alpha -1}}{x_{0}^{\alpha }[1+(x/x_{0})^{\alpha }]^{2}}},\qquad x>0.$

See also

• Generalized logistic distribution