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

The Champernowne distribution has a probability density function given by

f(y;α,λ,y0)=ncosh[α(yy0)]+λ,<y<,

where α,λ,y0 are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as

f(y)=n1/2eα(yy0)+λ+1/2eα(yy0),

using the fact that coshy=(ey+ey)/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 λ=1 it is the Burr Type XII density.

When y0=0,α=1,λ=1,

f(y)=1ey+2+ey=ey(1+ey)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)=nx[1/2(x/x0)α+λ+a/2(x/x0)α],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)=αxα1x0α[1+(x/x0)α]2,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 (4): 591–614. doi:10.2307/1907644. 
  3. Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal 63 (250): 318–351. doi:10.2307/2227127. 
  4. Fisk, P. R. (1961). "The graduation of income distributions". Econometrica 29 (2): 171–185. doi:10.2307/1909287.