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

$f(y;\alpha ,\lambda ,y_0)=\frac{n}{\cosh [\alpha (y-y_0)]+\lambda },-\mathrm{\infty }<y<\mathrm{\infty },$

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

$f(y)=\frac{n}{\frac{1}{2}{e}^{\alpha (y-y_0)}+\lambda +\frac{1}{2}{e}^{-\alpha (y-y_0)}},$

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

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 $\lambda =0$ ($\alpha =\frac{\pi }{2},y_0=0$) 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

$f(x)=\frac{n}{x[1/2(x/x_0{)}^{-\alpha }+\lambda +a/2(x/x_0{)}^{\alpha }]},x>0,$

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

$f(x)=\frac{\alpha x^{\alpha -1}}{x_0^{\alpha }[1+(x/x_0{)}^{\alpha }{]}^2},x>0.$

See also

• Generalized logistic distribution