Champernowne distribution

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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[edit]

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[edit]

The density f(y) defines a symmetric distribution with median y0, which has tails somewhat heavier than a normal distribution.

Special cases[edit]

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[edit]

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[edit]

References[edit]

  1. ^ a b C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley.  Section 7.3 "Champernowne Distribution."
  2. ^ a b 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.