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

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

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

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

using the fact that $\cosh y = (e^y + e^{-y})/2.$

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 $\lambda=1$ 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^[1]

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

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

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

See also

• Generalized logistic distribution

References

1. C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."
2. Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica 20: 591–614. doi:10.2307/1907644. JSTOR 1907644.
3. Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal 63 (250): 318–351. doi:10.2307/2227127. JSTOR 2227127.
4. Fisk, P. R. (1961). "The graduation of income distributions". Econometrica 29: 171–185. doi:10.2307/1909287.