Pareto distribution

Pareto

Probability density function Pareto probability density functions for various k  with xm = 1. The horizontal axis is the x  parameter. As k → ∞ the distribution approaches δ(x − xm) where δ is the Dirac delta function.
Cumulative distribution function Image:Pareto distributionCDF.png Pareto cumulative distribution functions for various k  with xm = 1. The horizontal axis is the x  parameter.
Parameters	Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m}>0\, scale (real) Failed to parse (Missing texvc executable; please see math/README to configure.): k>0\, shape (real)
Support	Failed to parse (Missing texvc executable; please see math/README to configure.): x \in [x_\mathrm{m}; +\infty)\!
Probability density function (pdf)	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{k\,x_\mathrm{m}^k}{x^{k+1}}\!
Cumulative distribution function (cdf)	Failed to parse (Missing texvc executable; please see math/README to configure.): 1-\left(\frac{x_\mathrm{m}}{x}\right)^k\!
Mean	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{k\,x_\mathrm{m}}{k-1}\! for Failed to parse (Missing texvc executable; please see math/README to configure.): k>1
Median	Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m} \sqrt[k]{2}
Mode	Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m}\,
Variance	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x_\mathrm{m}^2k}{(k-1)^2(k-2)}\! for Failed to parse (Missing texvc executable; please see math/README to configure.): k>2
Skewness	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2(1+k)}{k-3}\,\sqrt{\frac{k-2}{k}}\! for Failed to parse (Missing texvc executable; please see math/README to configure.): k>3
Excess kurtosis	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{6(k^3+k^2-6k-2)}{k(k-3)(k-4)}\! for Failed to parse (Missing texvc executable; please see math/README to configure.): k>4
Entropy	Failed to parse (Missing texvc executable; please see math/README to configure.): \ln\left(\frac{k}{x_\mathrm{m}}\right) - \frac{1}{k} - 1\!
Moment-generating function (mgf)	undefined; see text for raw moments
Characteristic function	Failed to parse (Missing texvc executable; please see math/README to configure.): k(-ix_\mathrm{m}t)^k\Gamma(-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 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^[1]. It can be seen from the probability density function (PDF) graph on the right, that the "probability" or fraction of the population Failed to parse (Missing texvc executable; please see math/README to configure.): 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 (a few words are used often, lots of words are used infrequently)
• 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Pr(X>x)=\left(\frac{x}{x_\mathrm{m}}\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.

Density function

It follows that the probability density function is

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): E(X)=\frac{kx_\mathrm{m}}{k-1} \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{var}(X)=\left(\frac{x_\mathrm{m}}{k-1}\right)^2 \frac{k}{k-2}.

(If Failed to parse (Missing texvc executable; please see math/README to configure.): k \le 2 , the variance is infinite). The raw moments are found to be

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_n'=\frac{kx_\mathrm{m}^n}{k-n}, \,

but they are only defined for Failed to parse (Missing texvc executable; please see math/README to configure.): k>n . This means that the moment generating function, which is just a Taylor series in Failed to parse (Missing texvc executable; please see math/README to configure.): x

with Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_n'/n!
as coefficients, is not defined. The characteristic function is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi(t;k,x_\mathrm{m})=k(-ix_\mathrm{m} t)^k\Gamma(-k,-ix_\mathrm{m} t),

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{k\rightarrow \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{ X₁, ..., X_n } and (X₁ + ... + X_n)/min{ X₁, ..., 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 wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF Failed to parse (Missing texvc executable; please see math/README to configure.): (f(x))

or the CDF Failed to parse (Missing texvc executable; please see math/README to configure.): (F(x))
as

Failed to parse (Missing texvc executable; please see math/README to configure.): 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')\,dF'}{\int_0^1 x(F')\,dF'}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): x(F)=\frac{x_\mathrm{m}}{(1-F)^{1/k}}

and the Lorenz curve is calculated to be

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): G = 1-2\int_0^1L(F)\,dF = \frac{1}{2k-1}

(see Aaberge 2005).

Parameter estimation

The likelihood function for the Pareto distribution parameters k and Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m} , given a sample Failed to parse (Missing texvc executable; please see math/README to configure.): x = (x_1, x_2, \dots, x_n) , is

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \ell(k, x_\mathrm{m})

is monotonically increasing with Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m}

, that is, the greater the value of Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m} , the greater the value of the likelihood function. Hence, since Failed to parse (Missing texvc executable; please see math/README to configure.): x \ge x_\mathrm{m} , we conclude that

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat k = \frac n {\sum _i {\left( \ln x_i - \ln \widehat x_\mathrm{m} \right)}}.

The expected statistical error is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma = \frac {\widehat k} {\sqrt n}.

[1]

Graphical representation

The charateristic 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 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle k_\mathrm{Gamma}=k_\mathrm{Pareto}\,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_\mathrm{Gamma}=\frac1{x_{\mathrm{m}_\mathrm{Pareto}}}.

This process generates data starting at 0, so then we need to add Failed to parse (Missing texvc executable; please see math/README to configure.): x_\mathrm{m} .

Alternatively, random samples can be generated using inverse transform sampling. Given a random variate Failed to parse (Missing texvc executable; please see math/README to configure.): U

drawn from the uniform distribution on the unit interval Failed to parse (Missing texvc executable; please see math/README to configure.): (0; 1)

, the variate

Failed to parse (Missing texvc executable; please see math/README to configure.): T=\frac{x_\mathrm{m}}{U^{1/k}}

is Pareto-distributed. [2]

Generalized Pareto distribution

The family of generalized Pareto distributions (GPD) has three parameters Failed to parse (Missing texvc executable; please see math/README to configure.): \mu,\sigma \,

and Failed to parse (Missing texvc executable; please see math/README to configure.):  \xi \,

.

location (real)

scale (real)

shape (real)

The cumulative distribution function is

Failed to parse (Missing texvc executable; please see math/README to configure.): F_{(\xi,\mu,\sigma)}(x) = 1 - \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}

for Failed to parse (Missing texvc executable; please see math/README to configure.): x \geqslant \mu , and Failed to parse (Missing texvc executable; please see math/README to configure.): x \leqslant \mu - \sigma /\xi

 when Failed to parse (Missing texvc executable; please see math/README to configure.):  \xi < 0 \,
, where Failed to parse (Missing texvc executable; please see math/README to configure.): \mu\in\mathbb R
is the location parameter, Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma>0 \,
the scale parameter and Failed to parse (Missing texvc executable; please see math/README to configure.): \xi\in\mathbb R
the shape parameter.  Note that some references give the "shape parameter" as Failed to parse (Missing texvc executable; please see math/README to configure.):  \kappa =  - \xi \,

.

The probability density function is

Failed to parse (Missing texvc executable; please see math/README to configure.): f_{(\xi,\mu,\sigma)}(x) = \frac{1}{\sigma}\left(1 + \frac{\xi (x-\mu)}{\sigma}\right)^{\left(-\frac{1}{\xi} - 1\right)}.

again, for Failed to parse (Missing texvc executable; please see math/README to configure.): x \geqslant \mu , and Failed to parse (Missing texvc executable; please see math/README to configure.): x \leqslant \mu - \sigma /\xi

 when Failed to parse (Missing texvc executable; please see math/README to configure.):  \xi < 0 \,
.

Generating generalized Pareto random variables

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

Failed to parse (Missing texvc executable; please see math/README to configure.): X = \mu + \frac{\sigma (U^{-\xi}-1)}{\xi} \sim \mbox{GPD}(\mu,\sigma,\xi).

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Annotations

1. ^ For a two-quantile population, where 18% of the population owns 82% of the wealth, the Theil index takes the value 1.

References

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

See also

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

External links

• 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
• Code to generate Pareto-distributed variables.
• The SOCR Resource provides interactive interface to Pareto distribution and Pareto sampling and simulation.

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