Pareto distribution

Pareto Type I

Probability density function Pareto Type I probability density functions for various α (labeled "k") with xm = 1. The horizontal axis is the x  parameter. As α → ∞ the distribution approaches δ(x − xm) where δ is the Dirac delta function.
Cumulative distribution function Pareto Type I cumulative distribution functions for various α(labeled "k") with xm = 1. The horizontal axis is the x  parameter.
Parameters	$x_\mathrm{m}>0\,$ scale (real) $\alpha>0\,$ shape (real)
Support	$x \in [x_\mathrm{m}, +\infty)\!$
PDF	$\frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m\!$
CDF	$1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha \text{ for } x \ge x_m\!$
Mean	$\frac{\alpha\,x_\mathrm{m}}{\alpha-1}\text{ for }\alpha>1\,$
Median	$x_\mathrm{m} \sqrt[\alpha]{2}$
Mode	$x_\mathrm{m}\,$
Variance	$\frac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)}\text{ for }\alpha>2\,$
Skewness	$\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3\,$
Ex. kurtosis	$\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4\,$
Entropy	$\ln\left(\frac{x_\mathrm{m}}{\alpha}\right) + \frac{1}{\alpha} + 1\!$
MGF	$\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0\,$
CF	$\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)\,$
Fisher information	$\begin{pmatrix}\frac{\alpha}{x_m^2} & -\frac{1}{x_m} \\ -\frac{1}{x_m} & \frac{1}{\alpha^2}\end{pmatrix}$

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 sometimes referred to as the Bradford distribution.

Definition

If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

$\overline{F}(x) = \Pr(X>x) = \begin{cases} \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & \text{for }x\ge x_\mathrm{m}, \\ 1 & \text{for } x < x_\mathrm{m}. \end{cases}$

where x_m is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter x_m and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.

Properties

Cumulative distribution function

From the definition, the cumulative distribution function of a Pareto random variable with parameters α and x_m is

$F_X(x) = \begin{cases} 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & \text{for } x \ge x_\mathrm{m}, \\ 0 & \text{for }x < x_\mathrm{m}. \end{cases}$

When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log-log plot, the distribution is represented by a straight line.

Probability density function

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

$f_X(x)= \begin{cases} \alpha\,\dfrac{x_\mathrm{m}^\alpha}{x^{\alpha+1}} & \text{for }x > x_\mathrm{m}, \\[12pt] 0 & \text{for } x < x_\mathrm{m}. \end{cases}$

Moments and characteristic function

• The expected value of a random variable following a Pareto distribution with α > 1 is

$E(X)=\frac{\alpha x_\mathrm{m}}{\alpha-1} \,$

(if α ≤ 1, the expected value does not exist).

• The variance of a random variable following a Pareto distribution with α > 2 is

$\mathrm{Var}(X)=\left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2}.$

(If α ≤ 2, the variance does not exist.)

• The raw moments are

$\mu_n'=\frac{\alpha x_\mathrm{m}^n}{\alpha-n}, \,$

but the nth moment exists only for n < α.

• The moment generating function is only defined for non-positive values t ≤ 0 as

$M\left(t,\alpha,x_\mathrm{m}\right) = E(e^{tX}) = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)\text{ and }M\left(0,\alpha,x_\mathrm{m}\right)=1.\,$

• The characteristic function is given by

$\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),$

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

Degenerate case

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

$\lim_{\alpha\rightarrow \infty} f(x;\alpha,x_\mathrm{m})=\delta(x-x_\mathrm{m}). \,$

Conditional distributions

The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number x₁ exceeding x_m, is a Pareto distribution with the same Pareto index α but with minimum x₁ instead of x_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 [x_m, ∞) for some x_m > 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.^{[citation needed]}

Generalized Pareto distributions

See also: Generalized Pareto distribution

There is a hierarchy of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions. Pareto Type IV contains Pareto Type I and II as special cases. The Feller–Pareto distribution generalizes Pareto Type IV.

Pareto Types I–IV

The Pareto distribution hierarchy is summarized in the table comparing the survival functions (complementary CDF). The Pareto distribution of the second kind is also known as the Lomax distribution.

Pareto Distributions

	$\overline{F}(x)=1-F(x)$	Support	Parameters
Type I	$\left[\frac{x}{\sigma}\right]^{-\alpha}$	$x > \sigma$	$\alpha,\sigma > 0$
Type II	$\left[1 + \frac{x-\mu}{\sigma}\right]^{-\alpha}$	$x > \mu$	$\alpha,\sigma > 0, \mu \in \mathbb R$
Lomax	$\frac{C^\alpha}{(x+C)^\alpha}$	$x \geq 0$	$\alpha,C>0$
Type III	$\left[1 + \left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-1}$	$x > \mu$	$\sigma, \gamma>0, \mu \in \mathbb R$
Type IV	$\left[1 + \left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-\alpha}$	$x > \mu$	$\alpha, \sigma, \gamma > 0, \mu \in \mathbb R$

The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are:

$P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),$ and

$P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha).$

Existence of the mean, and variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments exist for some δ>0, as shown in the table below, where δ is not necessarily an integer.

Moments of Pareto I–IV Distributions (case μ=0)

	$E[X]$	Condition	$E[X^\delta]$	Condition
Type I	$\frac{\sigma \alpha}{\alpha-1}$	$\alpha > 1$	$\frac{\sigma^\delta \alpha}{\alpha-\delta}$	$\delta < \alpha$
Type II	$\frac{ \sigma }{\alpha-1}$	$\alpha > 1$	$\frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}$	$-1 < \delta < \alpha$
Type III	$\sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)$	$-1<\gamma<1$	$\sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)$	$-\gamma^{-1}<\delta<\gamma^{-1}$
Type IV	$\frac{\sigma\Gamma(\alpha-\gamma \alpha)\Gamma(1+\gamma)}{\Gamma(\alpha)}$	$-1<\gamma<\alpha$	$\frac{\sigma^\delta\Gamma(\alpha-\gamma \alpha)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}$	$-\gamma^{-1}<\delta<\alpha/\gamma$

Feller–Pareto distribution

Feller defines a Pareto variable by transformation U = Y⁻¹ − 1 of a beta random variable Y, whose probability density function is

$f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0<y<1; \gamma_1,\gamma_2>0,$

where B( ) is the beta function. If

$W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma>0, \gamma>0,$

then W has a Feller–Pareto distribution FP(μ, σ, γ, γ₁, γ₂).

If $U_1 \sim \Gamma(\delta_1, 1)$ and $U_2 \sim \Gamma(\delta_2, 1)$ are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is

$W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma$

and we write W ~ FP(μ, σ, γ, δ₁, δ₂). Special cases of the Feller–Pareto distribution are

• $FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)$
• $FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)$
• $FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)$
• $FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha)$

Applications

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. He also used it to describe distribution of income. 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. However, the 80-20 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows 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, 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)
• Hard disk drive error rates
• Clusters of Bose–Einstein condensate near absolute zero^{[citation needed]}
• 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)^{[citation needed]}
• The standardized price returns on individual stocks

Fitted cumulative Pareto distribution to maximum one-day rainfalls

• 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)^{[citation needed]}
• Areas burnt in forest fires
• Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.
• In hydrology the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Relation to other distributions

Relation to the exponential distribution

The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum x_m and index α, then

$Y = \log\left(\frac{X}{x_\mathrm{m}}\right).$

is exponentially distributed with intensity (rate parameter) α. Equivalently, if Y is exponentially distributed with intensity α, then

$x_\mathrm{m} e^Y \,$

is Pareto-distributed with minimum x_m and index α.

This can be shown using the standard change of variable techniques:

$\Pr(Y<y) = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) = \Pr(X<x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}.$

The last expression is the cumulative distribution function of an exponential distribution with intensity α.

Relation to the log-normal distribution

Note that the Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution. (Both of these latter two distributions are "basic" in the sense that the logarithms of their density functions are linear and quadratic, respectively, functions of the observed values.)^{[citation needed]}

Relation to the generalized Pareto distribution

The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.

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.

Relation to the "Pareto principle"

The "80-20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log₄(5)=log(5)/log(4), approximately 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown to be mathematically equivalent:

• Income is distributed according to a Pareto distribution with index α > 1.
• There is some number 0 ≤ p ≤ 1/2 such that 100p% of all people receive 100(1 − p)% of all income, and similarly for every real (not necessarily integer) n > 0, 100pⁿ% of all people receive 100(1 − p)ⁿ% of all income.

This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.

This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.

Lorenz curve and Gini coefficient

Lorenz curves for a number of Pareto distributions. The case α = ∞ corresponds to perfectly equal distribution (G = 0) and the α = 1 line corresponds to complete inequality (G = 1)

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 ƒ or the CDF F as

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

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

$x(F)=\frac{x_\mathrm{m}}{(1-F)^{1/\alpha}}$

and the Lorenz curve is calculated to be

$L(F) = 1-(1-F)^{1-1/\alpha},\,$

where α 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 (α = ∞) 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

$G = 1-2\int_0^1L(F)\,dF = \frac{1}{2\alpha-1}$

(see Aaberge 2005).

Parameter estimation

The likelihood function for the Pareto distribution parameters α and x_m, given a sample x = (x₁, x₂, ..., x_n), is

$L(\alpha, x_\mathrm{m}) = \prod _{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod _{i=1}^n \frac 1 {x_i^{\alpha+1}}. \!$

Therefore, the logarithmic likelihood function is

$\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum _{i=1} ^n \ln x_i. \!$

It can be seen that $\ell(\alpha, x_\mathrm{m})$ 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 α, we compute the corresponding partial derivative and determine where it is zero:

$\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.$

Thus the maximum likelihood estimator for α is:

$\widehat \alpha = \frac{n}{\sum _i \left( \ln x_i - \ln \widehat x_\mathrm{m} \right)}.$

The expected statistical error is:

$\sigma = \frac {\widehat \alpha} {\sqrt n}.$

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.^{[citation needed]}

Random sample generation

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 given by

$T=\frac{x_\mathrm{m}}{U^{1/\alpha}}$

is Pareto-distributed.^{[citation needed]} If U is uniformly distributed on [0, 1), it can be exchanged for (1 - U).

Variants

Bounded Pareto distribution

See also: Truncated distribution

Bounded Pareto

Parameters	$L > 0 \,$ location (real) $H > L \,$ location (real) $\alpha > 0 \,$ shape (real)
Support	$L \leqslant x \leqslant H \,$
PDF	$\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}$
CDF	$\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}$
Mean	$\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1$
Median	$L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}$
Variance	$\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2$

The bounded (or truncated) Pareto distribution has three parameters α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value. (The variance in the table on the right should be interpreted as 2nd moment).

The probability density function is

$\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}$

where L ≤ x ≤ H, and α > 0.

Generating bounded Pareto random variables

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

$\left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}$

is bounded Pareto-distributed.^{[citation needed]}

Symmetric Pareto distribution

The symmetric Pareto distribution can be defined by the probability density function:

$f(x;\alpha,x_\mathrm{m}) = \begin{cases} (\alpha x_\mathrm{m}^\alpha/2) |x|^{-\alpha-1} & \text{for }|x|>x_\mathrm{m} \\ 0 & \text{otherwise}. \end{cases}$

It has a similar shape to a Pareto distribution for $x > x_\mathrm{m}$ and is mirror symmetric about the vertical axis.

See also

• Bradford's law
• Pareto analysis
• Pareto efficiency
• Pareto interpolation
• Power law probability distributions
• Traffic generation model

References

• M. O. Lorenz (1905). "Methods of measuring the concentration of wealth". Publications of the American Statistical Association 9 (70): 209–219. Bibcode 1905PAmSA...9..209L. doi:10.2307/2276207. 
• Pareto V (1965) "La Courbe de la Repartition de la Richesse" (Originally published in 1896). In: Busino G, editor. Oevres Completes de Vilfredo Pareto. Geneva: Librairie Droz. pp. 1–5.
• Pareto, V. (1895). La legge della domanda. Giornale degli Economisti, 10, 59–68. English translation in Rivista di Politica Economica, 87 (1997), 691–700.
• Pareto, V. (1897). Cours d'économie politique. Lausanne: Ed. Rouge.

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
• syntraf1.c is a C program to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.
• "Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes" /Mark E. Crovella and Azer Bestavros
• Weisstein, Eric W., "Pareto distribution" from MathWorld.