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

References

  1. ↑ 1.0 1.1 C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."
  2. ↑ 2.0 2.1 Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica 20: 591–614. JSTOR 1907644.
  3. ↑ Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal 63 (250): 318–351. JSTOR 2227127.
  4. ↑ Fisk, P. R. (1961). "The graduation of income distributions". Econometrica, 29, 171–185.