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.^[1][2][3] Champernowne developed the distribution to describe the logarithm of income.^[2]

Definition

The Champernowne distribution has a probability density function given by

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

where ${\displaystyle \alpha ,\lambda ,y_{0}}$ are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as

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

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

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

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

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

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

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^[1]

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

where x₀ = exp(y₀) is the median income. If λ = 1, this distribution is often called the Fisk distribution,^[4] which has density

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

See also

• Generalized logistic distribution