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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. Champernowne developed the distribution to describe the logarithm of income.

Contents

Definition

The Champernowne distribution has a probability density function given by

f ( y ; α , λ , y 0 ) = n cosh ⁡ [ α ( y − y 0 ) ] + λ , − ∞ < y < ∞ ,

where α , λ , y 0 are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as

f ( y ) = n 1 / 2 e α ( y − y 0 ) + λ + 1 / 2 e − α ( 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 λ = 1 it is the Burr Type XII density.

When y 0 = 0 , α = 1 , λ = 1 ,

f ( y ) = 1 e y + 2 + e − y = 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 ) = n x [ 1 / 2 ( x / x 0 ) − α + λ + a / 2 ( x / x 0 ) α ] , 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 ) = α x α − 1 x 0 α [ 1 + ( x / x 0 ) α ] 2 , x > 0.

References

Champernowne distribution Wikipedia
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