Английская Википедия: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.

Definition

The Champernowne distribution has a probability density function given by

<math>

f(y;\alpha, \lambda, y_0 ) = \frac{n}{\cosh[\alpha(y - y_0)] + \lambda}, \qquad -\infty < y < \infty,

</math>

where <math> \alpha, \lambda, y_0</math> are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as

<math>

f(y) = \frac{n}{1/2 e^{\alpha(y-y_0)} + \lambda + 1/2 e^{-\alpha(y-y_0)}},

</math>

using the fact that <math> \cosh y = (e^y + e^{-y})/2.</math>

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 <math>\lambda=1</math> it is the Burr Type XII density.

When <math> y_0 = 0, \alpha=1, \lambda=1 </math>,

<math>

f(y) = \frac{1}{e^y + 2 + e^{-y}} = \frac{e^y}{(1+e^y)^2},

</math>

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

<math>

f(x) = \frac{n}{x [1/2(x/x_0)^{-\alpha} + \lambda + a/2(x/x_0)^\alpha ]}, \qquad x > 0,

</math>

where x₀ = exp(y₀) is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density

<math>

f(x) = \frac{\alpha x^{\alpha - 1}}{x_0^\alpha [1 + (x/x_0)^\alpha]^2}, \qquad x > 0.

</math>

See also

• Generalized logistic distribution