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 12 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. $$