Champernowne distribution

Last updated March 17, 2023

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;\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}{1/2e^{\alpha (y-y_{0})}+\lambda +1/2e^{-\alpha (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 $\lambda =1$ 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

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.

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References

1 2 C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."
1 2 Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644.
↑ Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal. 63 (250): 318–351. doi:10.2307/2227127. JSTOR 2227127.
↑ Fisk, P. R. (1961). "The graduation of income distributions". Econometrica. 29 (2): 171–185. doi:10.2307/1909287. JSTOR 1909287.

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[KK-1] 1 2 C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. Section 7.3 "Champernowne Distribution."

[Champ52-2] 1 2 Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644.

[Champ53-3] Champernowne, D. G. (1953). "A Model of Income Distribution". The Economic Journal. 63 (250): 318–351. doi:10.2307/2227127. JSTOR 2227127.

[4] Fisk, P. R. (1961). "The graduation of income distributions". Econometrica. 29 (2): 171–185. doi:10.2307/1909287. JSTOR 1909287.

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