\!| cdf =$1-\backslash left(\backslash frac\{x\_\backslash mathrm\{m\}\}\{x\}\backslash right)^k\backslash !$| mean =$\backslash frac\{k\backslash ,x\_\backslash mathrm\{m\}\}\{k-1\}\backslash !$ for $k>1$| median =$x\_\backslash mathrm\{m\}\; \backslash sqrt*\{2\}$| mode =$x\_\backslash mathrm\{m\}\backslash ,$| variance =$\backslash frac\{x\_\backslash mathrm\{m\}^2k\}\{(k-1)^2(k-2)\}\backslash !$ for $k>2$| skewness =$\backslash frac\{2(1+k)\}\{k-3\}\backslash ,\backslash sqrt\{\backslash frac\{k-2\}\{k\}\}\backslash !$ for $k>3$| kurtosis =$\backslash frac\{6(k^3+k^2-6k-2)\}\{k(k-3)(k-4)\}\backslash !$ for $k>4$| entropy =$\backslash ln\backslash left(\backslash frac\{k\}\{x\_\backslash mathrm\{m\}\}\backslash right)\; -\; \backslash frac\{1\}\{k\}\; -\; 1\backslash !$| mgf =undefined; see text for raw moments| char =$k(-ix\_\backslash mathrm\{m\}t)^k\backslash Gamma(-k,-ix\_\backslash mathrm\{m\}t)\backslash ,$| }} 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. Outside the field of economics it is at times referred to 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 probability density function (PDF) graph on the right, that the "probability" or fraction of the population $f(x)$ that owns a small amount of wealth per person (x) is rather high, and then decreases steadily as wealth increases. This distribution is not 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 sizes 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 values 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
• Sizes of sand particles
• Sizes of meteorites
• Numbers of species per genus (There is 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

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If X is a random variable with a Pareto distribution, then the probability that X is greater than some number x is given by

$\backslash Pr(X>x)=\backslash left(\backslash frac\{x\}\{x\_\backslash mathrm\{m\}\}\backslash right)^\{-k\}$

for all x ≥ x_m, where x_m 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. When this distribution is used to model the distribution of wealth, then the parameter k is called the Pareto index.

It follows that the probability density function is

$f(x;k,x\_\backslash mathrm\{m\})\; =\; k\backslash ,\backslash frac\{x\_\backslash mathrm\{m\}^k\}\{x^\{k+1\}\}\backslash \; \backslash mbox\{for\}\backslash \; x\; \backslash ge\; x\_\backslash mathrm\{m\}.\; \backslash ,$

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

$E(X)=\backslash frac\{kx\_\backslash mathrm\{m\}\}\{k-1\}\; \backslash ,$

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

$\backslash mathrm\{var\}(X)=\backslash left(\backslash frac\{x\_\backslash mathrm\{m\}\}\{k-1\}\backslash right)^2\; \backslash frac\{k\}\{k-2\}.$

(If $k\; \backslash le\; 2$, the variance is infinite). The raw moments are found to be

$\backslash mu\_n\text{'}=\backslash frac\{kx\_\backslash mathrm\{m\}^n\}\{k-n\},\; \backslash ,$

but they are only defined for $k>n$. This means that the moment generating function, which is just a Taylor series in $x$ with $\backslash mu\_n\text{'}/n!$ as coefficients, is not defined. The characteristic function is given by

$\backslash varphi(t;k,x\_\backslash mathrm\{m\})=k(-ix\_\backslash mathrm\{m\}\; t)^k\backslash Gamma(-k,-ix\_\backslash mathrm\{m\}\; t),$

where Γ(a,x) is the incomplete Gamma function. The Pareto distribution is related to the exponential distribution by

$f(x;k,x\_\backslash mathrm\{m\})=\backslash mathrm\{Exponential\}(\backslash ln(x/x\_\backslash mathrm\{m\});k).\backslash ,$

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

$\backslash lim\_\{k\backslash rightarrow\; \backslash infty\}\; f(x;k,x\_\backslash mathrm\{m\})=\backslash delta(x-x\_\backslash mathrm\{m\}).\; \backslash ,$

Pareto, Lorenz, and Gini

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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 $(f(x))$ or the CDF $(F(x))$ as

$L(F)=\backslash frac\{\backslash int\_\{x\_\backslash mathrm\{m\}\}^\{x(F)\}$

xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}

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

$x(F)=\backslash frac\{x\_\backslash mathrm\{m\}\}\{(1-F)^\{1/k\}\}$

and the Lorenz curve is calculated to be

$L(F)\; =\; 1-(1-F)^\{1-1/k\},\backslash ,$

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 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

$G\; =\; 1-2\backslash int\_0^1L(F)\backslash ,dF\; =\; \backslash frac\{1\}\{2k-1\}$

(see Aaberge 2005).

Parameter estimation

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The likelihood function for the Pareto distribution parameters k and $x\_\backslash mathrm\{m\}$, given a sample $x\; =\; (x\_1,\; x\_2,\; \backslash dots,\; x\_n)$, is

$L(k,\; x\_\backslash mathrm\{m\})\; =\; \backslash prod\; \_\{i=1\}\; ^n\; \{k\; \backslash frac\; \{x\_\backslash mathrm\{m\}^k\}\; \{x\_i^\{k+1\}\}\}\; =\; k^n\; x\_\backslash mathrm\{m\}^\{nk\}\; \backslash prod\; \_\{i=1\}\; ^n\; \{\backslash frac\; 1\; \{x\_i^\{k+1\}\}\}.\; \backslash !$

Therefore, the logarithmic likelihood function is

$\backslash ell(k,\; x\_\backslash mathrm\{m\})\; =\; n\; \backslash ln\; k\; +\; nk\; \backslash ln\; x\_\backslash mathrm\{m\}\; -\; (k\; +\; 1)\; \backslash sum\; \_\{i=1\}\; ^n\; \{\backslash ln\; x\_i\}.\; \backslash !$

It can be seen that $\backslash ell(k,\; x\_\backslash mathrm\{m\})$ is monotonically increasing with $x\_\backslash mathrm\{m\}$, that is, the greater the value of $x\_\backslash mathrm\{m\}$, the greater the value of the likelihood function. Hence, since $x\; \backslash ge\; x\_\backslash mathrm\{m\}$, we conclude that

$\backslash widehat\; x\_\backslash mathrm\{m\}\; =\; \backslash min\; \_i\; \{x\_i\}.$

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

$\backslash frac\{\backslash partial\; \backslash ell\}\{\backslash partial\; k\}\; =\; \backslash frac\{n\}\{k\}\; +\; n\; \backslash ln\; x\_\backslash mathrm\{m\}\; -\; \backslash sum\; \_\{i=1\}\; ^n\; \{\backslash ln\; x\_i\}\; =\; 0.$

Thus the maximum likelihood estimator for k is:

$\backslash widehat\; k\; =\; \backslash frac\; n\; \{\backslash sum\; \_i\; \{\backslash left(\; \backslash ln\; x\_i\; -\; \backslash ln\; \backslash widehat\; x\_\backslash mathrm\{m\}\; \backslash right)\}\}.$

Generating a random sample from Pareto distribution

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Pareto distribution is not yet recognized by many programming languages. In actuarial field, Pareto distribution in widely used to estimate portfolio costs. As a matter of fact, it can be quite demanding to get data from this particular probability distribution. One can easily generate a random sample from Pareto distribution by mixing two random variables, which are usually built-in in many Statistical tools. The process is quite simple; one has to generate numbers from an exponential distribution with its lambda equal to randomly generated sample from a gamma distribution, with alpha and lambda respectively equal to the arguments of the initial Pareto distribution. Data generated from this process won't be translated from the origin to $x\_\backslash mathrm\{m\}$.

Example using R

This code will generate n numbers from a Pareto distribution, with parameters $x\_\backslash mathrm\{m\}$ = lambda and $k$ = alpha.
In this very specific example, these values will be set to

• n = 10
• alpha = 20
• lambda = 10

These are arbitrary values, used to show working R code

n <- 10 # number of randomly generated numbers
alpha <- 20 # Shape parameter of Pareto distribution
lambda <- 10 # Location parameter of Pareto distribution
X <- rexp(n,rgamma(n,shape=alpha,rate=lambda))

References

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

See also

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

External links

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• The Pareto, Zipf and other power laws / William J. Reed -- PDF
• Gini's Nuclear Family / Rolf Aabergé. -- In: International Conference to Honor Two Eminent Social Scientists, May, 2005 -- PDF

Continuous distributions | Power laws | Socioeconomics

Pareto-Verteilung | Distribución Pareto | Distribution de Pareto | Variabile casuale paretiana | Закон Парето | Pareto-jakauma | Распределение Парето | 帕累托分布