Pareto distribution

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[1] (Italian: [] pə-RAY-toh),[2] is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena. Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.[3][4] The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value (α) of log45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena[5] and human activities.[6][7]

If X is a random variable with a Pareto (Type I) distribution,[8] then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

where xm 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 xm 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.

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

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

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.

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: It follows from the formula for the probability density function that for xxm,

There is a hierarchy [8][12] of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.[8][12][13] Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto[12][14] distribution generalizes Pareto Type IV.

The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).

When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.[15]

In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.

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

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

Feller[12][14] defines a Pareto variable by transformation U = Y−1 − 1 of a beta random variable Y, whose probability density function is

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

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

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

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

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. (See the previous section.)

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.

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.

If U is uniformly distributed on (0, 1), then applying inverse-transform method [18]

The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.[19]

The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:[19]

This distribution has two parameters: a and b. It is symmetric by b. Then the mathematic expectation is b. When, it has variance as following:

The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:

This distribution is symmetric by zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.[19]

The univariate Pareto distribution has been extended to a multivariate Pareto distribution.[20]

The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1x2, ..., xn), is

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

Vilfredo 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.[4] 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.[23] 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.[citation needed] 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. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, net worth may even be negative.) 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:

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 an infinite expected value, and so cannot reasonably model income distribution.

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 or the CDF F as

The solution is that α equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.[34]

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

is Pareto-distributed.[35] If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).