Pareto distribution

Template:Probability distribution The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. This distribution is also known, mostly outside economics, as the Bradford distribution.

Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population owns 80% of the wealth. It can be seen from the PDF graph on the right, that the "probability" or fraction of the population p(x)  that owns a small amount of wealth per person (x ) is rather high, and then decreases steadily as wealth increases. This distribution is not just limited to describing wealth or income distribution, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:

• Frequencies of words in longer texts
• The size of human settlements (few cities, many hamlets/villages)
• File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)
• Clusters of Bose-Einstein condensate near absolute zero
• The value of oil reserves in oil fields (a few large fields, many small fields)
• The length distribution in jobs assigned supercomputers (a few large ones, many small ones)
• The standardized price returns on individual stocks
• Size of sand particles
• Size of meteorites
• Number of species per genus (please note the subjectivity involved: The tendency to divide a genus into two or more increases with the number of species in it)
• Areas burnt in forest fires

Properties

Mathematically speaking, if X is a random variable with a Pareto distribution, then the probability distribution of X is characterized by the statement

<math>\Pr(X>x)=\left(\frac{x}{x_m}\right)^{-k}<math>

where x is any number greater than x_m, which is the (necessarily positive) minimum possible value of X, and k is a positive parameter. The family of Pareto distributions is parameterized by two quantities, x_m and k. The probability density is then

<math>p(x|k,x_m) = k\,\frac{x_m^k}{x^{k+1}}\ \mbox{for}\ x \ge x_m. \, <math>

Pareto distributions are continuous probability distributions. Zipf's law, also sometimes called the zeta distribution, may be thought of as a discrete counterpart of the Pareto distribution.

The expected value of a random variable following a Pareto distribution is

<math>x_m \; k \over k-1 \!<math>

(if k ≤ 1, the expected value is infinite). Its standard deviation is

<math>{x_m \over k-1} \sqrt{k \over k-2} \!<math>

(if k ≤ 2, the standard deviation is infinite).

The raw moments are found to be:

<math>\mu_n'=\frac{kx_m^n}{k-n} \!<math>

but they are only defined for <math>k>n<math>. This means that the moment generating function, which is just a Taylor series in <math>x<math> with <math>\mu_n'/n!<math> as coefficients, is not defined. The characteristic function is given by:

<math>\varphi(t|k,x_m)=k(-ix_mt)^k\Gamma(-k,-ix_mt)<math>

where Γ(a,x) is the incomplete Gamma function.

The Pareto distribution is related to the exponential distribution f(x|k) by:

<math>p(x|k,x_m)=f(\ln(x/x_m)|k)\,<math>

The Dirac delta function is a limiting case of the Pareto distribution:

<math>\lim_{k\rightarrow \infty} p(x|k,x_m)=\delta(x-x_m)<math>

Pareto, Lorenz, and Gini

The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F)  is written in terms of the PDF (p(x)) or the CDF (F(x)) as:

<math>L(F)=\frac{\int_{x_m}^{x(F)} xp(x)\,dx}{\int_{x_m}^\infty xp(x)\,dx}

=\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}<math>

where x(F) is the inverse of the CDF. For the Pareto distribution,

<math>x(F)=\frac{x_m}{(1-F)^{1/k}}<math>

and the Lorenz curve is calculated to be:

<math>L(F) = 1-(1-F)^{1-1/k}\,<math>

where k must be greater than or equal to unity, since the denominator in the expression for L(F)  is just the mean value of x . Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0,0] and [1,1], which is shown in black (k=∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated to be:

<math>G = 1-2\int_0^1L(F)dF = \frac{1}{2k-1}<math>

(see Aaberge 2005).

Parameter estimation

The likelihood function for the Pareto distribution parameters k and <math>x_m<math>, given a sample <math>x = (x_1, x_2, ..., x_n)<math>, is

<math>L(k, x_m) = \prod _{i=1} ^n {k \frac {x_m^k} {x_i^{k+1}}} = k^n x_m^{nk} \prod _{i=1} ^n {\frac 1 {x_i^{k+1}}}<math>.

Therefore, the logarithmic likelihood function is

<math>\ell(k, x_m) = n \ln k + nk \ln x_m - (k + 1) \sum _{i=1} ^n {\ln x_i}<math>.

It can be seen that <math>\ell(k, x_m)<math> is monotonically increasing with <math>x_m<math>, that is, the greater the value of <math>x_m<math>, the greater the value of the likelihood function. Hence, since <math>x \ge x_m<math>, we conclude that

<math>\hat x_m = \min _i {x_i}.<math>

To find the estimator for k, we compute the corresponding partial derivative and determine where it is zero:

<math>\frac {\partial \ell} {\partial k} = \frac n k + n \ln x_m - \sum _{i=1} ^n {\ln x_i} = 0<math>.

Thus the maximum likelihood estimators for k and <math>x_m<math> are:

<math>\hat x_m = \min _i {x_i}, \ \hat k = \frac n {\sum _i {\left( \ln x_i - \ln \hat x_m \right)}}<math>.

References

• Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statisical Association. 9: 209-219.

See also

• Pareto principle
• Pareto interpolation
• Pareto efficiency
• The Long Tail

External links

• William J. Reed: The Pareto, Zipf and other power laws, http://linkage.rockefeller.edu/wli/zipf/reed01_el.pdf

• Aaberge, Rolf, International Conference to Honor Two Eminent Social Scientists, May, 2005. Gini's Nuclear Family (http://www.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf)de:Pareto-Verteilung

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