Champernowne distribution

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

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.

References

References

1. C. Kleiber and S. Kotz. (2003). "Statistical Size Distributions in Economics and Actuarial Sciences". Wiley.
2. Champernowne, D. G.. (1952). "The graduation of income distributions". Econometrica.
3. (1961). "The graduation of income distributions". Econometrica.