Pareto distribution

Probability distribution
name =Pareto
type =density
pdf_

cdf_

Pareto cumulative distribution functions for various "k" with "x"_m = 1. The horizontal axis is the "x" parameter.
parameters =$x\_mathrm\{m\}>0,$ scale (real) $k>0,$ shape (real)
support =$x\; in\; [x\_mathrm\{m\};\; +infty)!$
pdf =$frac\{k,x\_mathrm\{m\}^k\}\{x^\{k+1!$
cdf =$1-left(frac\{x\_mathrm\{m\{x\}\; ight)^k!$
mean =$frac\{k,x\_mathrm\{m\{k-1\}!$ for $k>1$
median =$x\_mathrm\{m\}\; sqrt\; [k]\; \{2\}$
mode =$x\_mathrm\{m\},$
variance =$frac\{x\_mathrm\{m\}^2k\}\{(k-1)^2(k-2)\}!$ for $k>2$
skewness =$frac\{2(1+k)\}\{k-3\},sqrt\{frac\{k-2\}\{k!$ for $k>3$
kurtosis =$frac\{6(k^3+k^2-6k-2)\}\{k(k-3)(k-4)\}!$ for $k>4$
entropy =$lnleft(frac\{k\}\{x\_mathrm\{m\; ight)\; -\; frac\{1\}\{k\}\; -\; 1!$
mgf =undefined; "see text for raw moments"
char =$k(-ix\_mathrm\{m\}t)^kGamma(-k,-ix\_mathrm\{m\}t),$
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution that coincides with social, scientific, geophysical, actuarial, and many other types of observable phenomena. Outside the field of economics it is at times referred to as the Bradford distribution.

Pareto originally used this distribution to describe the allocation ofwealth among individuals since it seemed to show rather well the way thata larger portion of the wealth of any society is owned by a smaller percentageof 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 controls 80% of the wealth [For a two-quantile population, where 18% of the population owns 82% of the wealth, the Theil index takes the value 1.] . It can be seen from the probability density function (PDF) graph on the right, that the "probability" or fraction of the population that owns a small amount of wealth per person 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:

* 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
* Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.

Properties

Definition

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

:$Pr(X>x)=left(frac\{x\}\{x\_mathrm\{m\; ight)^\{-k\}$

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.

Density function

It follows (by differentation) that the probability density function is

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

Various properties

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

:$E(X)=frac\{kx\_mathrm\{m\{k-1\}\; ,$

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

:$mathrm\{var\}(X)=left(frac\{x\_mathrm\{m\{k-1\}\; ight)^2\; frac\{k\}\{k-2\}.$

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

:$mu\_n\text{'}=frac\{kx\_mathrm\{m\}^n\}\{k-n\},\; ,$

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

:$varphi(t;k,x\_mathrm\{m\})=k(-ix\_mathrm\{m\}\; t)^kGamma(-k,-ix\_mathrm\{m\}\; t),$

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

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

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

:$lim\_\{k\; ightarrow\; infty\}\; f(x;k,x\_mathrm\{m\})=delta(x-x\_mathrm\{m\}).\; ,$

A characterization theorem

Suppose "X"_"i", "i" = 1, 2, 3, ... are independent identically distributed random variables whose probability distribution is supported on the interval ["k", ∞) for some "k" > 0. Suppose that for all "n", the two random variables min{ "X"₁, ..., "X"_"n" } and ("X"₁ + ... + "X"_"n")/min{ "X"₁, ..., "X"_"n" } are independent. Then the common distribution is a Pareto distribution.

Relation to Zipf's law

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.

Pareto, Lorenz, and Gini

The Lorenz curve is often used to characterize income and wealthdistributions. For any distribution, the Lorenz curve "L"("F") is writtenin terms of the PDF ƒ or the CDF "F" as

:$L(F)=frac\{int\_\{x\_mathrm\{m^\{x(F)\}xf(x),dx\}\{int\_\{x\_mathrm\{m^infty\; xf(x),dx\}=frac\{int\_0^F\; x(F\text{'}),dF\text{'}\}\{int\_0^1\; x(F\text{'}),dF\text{'}\}$

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

:$x(F)=frac\{x\_mathrm\{m\{(1-F)^\{1/k$

and the Lorenz curve is calculated to be

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

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

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

(see Aaberge 2005).

Parameter estimation

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

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

Therefore, the logarithmic likelihood function is

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

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

: $widehat\; x\_mathrm\{m\}\; =\; min\; \_i\; \{x\_i\}.$

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

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

Thus the maximum likelihood estimator for "k" is:

: $widehat\; k\; =\; frac\; n\; \{sum\; \_i\; \{left(\; ln\; x\_i\; -\; ln\; widehat\; x\_mathrm\{m\}\; ight).$

The expected statistical error is:

: $sigma\; =\; frac\; \{widehat\; k\}\; \{sqrt\; n\}.$ [http://arxiv.org/abs/cond-mat/0412004v3]

Graphical representation

The characteristic curved 'long tail' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a log-log graph, which then takes the form of a straight line with negative gradient.

Generating a random sample from Pareto distribution

The Pareto distribution is not yet recognized by many programming languages. In the actuarial field, the Pareto distribution is 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 many statistical tools. The process is quite simple; one has to generate numbers from an exponential distribution with its λ equal to a random generated sample from a gamma distribution

: $displaystyle\; k\_mathrm\{Gamma\}=k\_mathrm\{Pareto\},$

and

: $heta\_mathrm\{Gamma\}=frac1\{x\_\{mathrm\{m\}\_mathrm\{Pareto\}.$

This process generates data starting at 0, so then we need to add $x\_mathrm\{m\}$.

Alternatively, random samples can be generated using inverse transform sampling. Given a random variate $U$ drawn from the uniform distribution on the unit interval $(0;\; 1)$, the variate

:$T=frac\{x\_mathrm\{m\{U^\{1/k$

is Pareto-distributed. [http://www.stat.psu.edu/~dhunter/R/2006mle.html]

Generalized Pareto distribution

The family of generalized Pareto distributions (GPD) has three parameters $mu,sigma\; ,$ and $xi\; ,$.

Probability distribution
name =Generalized Pareto
type =density
pdf_

cdf_

parameters =$mu\; in\; (-infty,infty)\; ,$ location (real)
$sigma\; in\; (0,infty)\; ,$ scale (real)
$xiin\; (-infty,infty)\; ,$ shape (real)
support =$x\; geqslant\; mu,;(xi\; geqslant\; 0)$

pdf =$frac\{1\}\{sigma\}(1\; +\; xi\; z\; )^\{-(1/xi\; +1)\}$
where $z=frac\{x-mu\}\{sigma\}$
cdf =$1-(1+xi\; z)^\{-1/xi\}\; ,$
mean =
median =$mu\; +\; frac\{sigma(\; 2^\{xi\}\; -1)\}\{xi\}$
mode =
variance =
skewness =
kurtosis =
entropy =
mgf =
char =

The cumulative distribution function is

: $F\_\{(xi,mu,sigma)\}(x)\; =\; 1\; -\; left(1+\; frac\{xi(x-mu)\}\{sigma\}\; ight)^\{-1/xi\}$

for $x\; geqslant\; mu$, and $x\; leqslant\; mu\; -\; sigma\; /xi$ when , where $muinmathbb\; R$ is the location parameter, $sigma>0\; ,$ the scale parameter and $xiinmathbb\; R$ the shape parameter. Note that some references give the "shape parameter" as $kappa\; =\; -\; xi\; ,$.

The probability density function is

: $f\_\{(xi,mu,sigma)\}(x)\; =\; frac\{1\}\{sigma\}left(1\; +\; frac\{xi\; (x-mu)\}\{sigma\}\; ight)^\{left(-frac\{1\}\{xi\}\; -\; 1\; ight)\}.$

again, for $x\; geqslant\; mu$, and $x\; leqslant\; mu\; -\; sigma\; /xi$ when .

Generating generalized Pareto random variables

If "U" is uniformly distributed on (0, 1] , then

:$X\; =\; mu\; +\; frac\{sigma\; (U^\{-xi\}-1)\}\{xi\}\; sim\; mbox\{GPD\}(mu,sigma,xi).$

Annotations

References

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

ee also

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

External links

* The Pareto, Zipf and other power laws / William J. Reed -- [http://linkage.rockefeller.edu/wli/zipf/reed01_el.pdf PDF]
* Gini's Nuclear Family / Rolf Aabergé. -- In: [http://www.unisi.it/eventi/GiniLorenz05/ International Conference to Honor Two Eminent Social Scientists] , May, 2005 -- [http://www.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf PDF]
* The SOCR Resource provides [http://socr.ucla.edu/htmls/SOCR_Distributions.html interactive interface to Pareto distribution] and [http://socr.ucla.edu/htmls/SOCR_Modeler.html Pareto sampling and simulation] .
* The CERN cern.jet.random package provides a [http://www.ssfnet.org/ssfImplementations.html Java implementation of the Pareto distribution] . Downloadable as part of the [http://www.ssfnet.org/ Scalable Simulation Framework] (SSF), among others.