Pareto distribution

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. 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:


 * 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.

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


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

for all x ≥ xm, where xm 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, xm 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


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

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


 * $$\mathrm{var}(X)=\left(\frac{x_\mathrm{m}}{k-1}\right)^2 \frac{k}{k-2}.$$

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


 * $$\mu_n'=\frac{kx_\mathrm{m}^n}{k-n}, \,$$

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


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

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


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

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


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

A characterization theorem
Suppose Xi, 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, ..., Xn } and (X1 + ... + Xn)/min{ X1, ..., Xn } 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 $$(f(x))$$ or the CDF $$(F(x))$$ 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,


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

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


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

(see Aaberge 2005).

Parameter estimation
The likelihood function for the Pareto distribution parameters k and $$x_\mathrm{m}$$, 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:


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

The expected statistical error is:


 * $$\sigma = \frac {\widehat k} {\sqrt n}. $$

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


 * $$\displaystyle k_\mathrm{Gamma}=k_\mathrm{Pareto}\, $$

and


 * $$\theta_\mathrm{Gamma}=\frac1{x_{\mathrm{m}_\mathrm{Pareto}}}.$$

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. 

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

The cumulative distribution function is


 * $$F_{(\xi,\mu,\sigma)}(x) = 1 - \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}$$

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

The probability density function is


 * $$f_{(\xi,\mu,\sigma)}(x) = \frac{1}{\sigma}\left(1 + \frac{\xi (x-\mu)}{\sigma}\right)^{\left(-\frac{1}{\xi} - 1\right)}.$$

again, for $$ x \geqslant \mu $$, and $$ x \leqslant \mu - \sigma /\xi $$ when $$ \xi < 0 \,$$.

Generating generalized Pareto random variables
If U is uniformly distributed on (0, 1 ], then


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