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
![{\displaystyle f(y;\alpha ,\lambda ,y_{0})={\frac {n}{\cosh[\alpha (y-y_{0})]+\lambda )),\qquad -\infty <y<\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3369724a8b7b0a49626894b3d9e5d5cd61d32d)
where
are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as
![{\displaystyle f(y)={\frac {n}{1/2e^{\alpha (y-y_{0})}+\lambda +1/2e^{-\alpha (y-y_{0})))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3b0b1ab9d870d24856afa50106608ec30c59a4)
using the fact that
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
it is the Burr Type XII density.
When
,
![{\displaystyle f(y)={\frac {1}{e^{y}+2+e^{-y))}={\frac {e^{y)){(1+e^{y})^{2))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3787d9e1924d193e433e065b72cd2b086bab41d0)
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]
![{\displaystyle f(x)={\frac {n}{x[1/2(x/x_{0})^{-\alpha }+\lambda +a/2(x/x_{0})^{\alpha }])),\qquad x>0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53e54175d2d90029329495593ecdb7043b149c7c)
where x0 = exp(y0) is the median income. If λ = 1, this distribution is often called the Fisk distribution,[4] which has density
![{\displaystyle f(x)={\frac {\alpha x^{\alpha -1)){x_{0}^{\alpha }[1+(x/x_{0})^{\alpha }]^{2))},\qquad x>0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8de85b10fc4eec441d19500ec826fe365311484e)