The Pareto distribution, named after the Italian polymath Vilfredo Pareto, is a probability distribution in the form of a power law that is used to describe social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle 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.
Empirical observation has shown that the Pareto distribution fits a wide range of cases, including natural phenomena and human activities.
The Pareto principle or "80:20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto. Pareto distributions with a shape value () of exhibit the Pareto principle.
Definitions 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
where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The type I Pareto distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. If this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
Cumulative distribution function From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is
Probability density function 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.
Properties Moments and characteristic function The expected value of a random variable following a Pareto distribution is The variance of a random variable following a Pareto distribution is (If α ≤ 1, the variance does not exist.) The raw moments are The moment generating function is only defined for non-positive values t ≤ 0 as Thus, since the expectation does not converge on an open interval containing we say that the moment generating function does not exist. The characteristic function is given by where Γ(a, x) is the incomplete gamma function.
The parameters may be solved for using the method of moments.S. Hussain, S.H. Bhatti (2018). Parameter estimation of Pareto distribution: Some modified moment estimators. Maejo International Journal of Science and Technology 12(1):11-27.
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 exceeding , is a Pareto distribution with the same Pareto index but with minimum instead of :
This implies that the conditional expected value (if it is finite, i.e. ) is proportional to :
In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called the Lindy effect or Lindy's Law.
A characterization theorem Suppose are independent identically distributed random variables whose probability distribution is supported on the interval for some . Suppose that for all , the two random variables and are independent. Then the common distribution is a Pareto distribution.
Geometric mean The geometric mean (G) isJohnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.
Harmonic mean The harmonic mean (H) is
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: It follows from the formula for the probability density function that for x ≥ xm,
Since α is positive, the gradient −(α + 1) is negative.
Related distributions Generalized Pareto distributions
There is a hierarchy Johnson, Kotz, and Balakrishnan (1994), (20.4). of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions. Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto "The densities (4.3) are sometimes called after the economist Pareto. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ Ax−α as x → ∞". distribution generalizes Pareto Type IV.
Pareto types I–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.
In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.
+Pareto distributions Support Parameters Type I Type II Lomax Type III Type IV
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.
+Moments of Pareto I–IV distributions (case μ = 0) Condition Condition Type I Type II Type III Type IV
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
where B( ) is the beta function. If
then W has a Feller–Pareto distribution FP(μ, σ, γ, γ1, γ2).
If and are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is
and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are
Inverse-Pareto Distribution / Power Distribution
When a random variable follows a pareto distribution, then its inverse follows a Power distribution. Inverse Pareto distribution is equivalent to a Power distribution
Relation to the exponential distribution 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
is Pareto-distributed with minimum xm and index α.
This can be shown using the standard change-of-variable techniques:
The last expression is the cumulative distribution function of an exponential distribution with rate α.
Pareto distribution can be constructed by hierarchical exponential distributions. section 5.3.1. Let and . Then we have and, as a result, .
More in general, if (shape-rate parametrization) and , then .
Equivalently, if and , then .
Relation to the log-normal distribution 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.)
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.
The Pareto distribution with scale and shape is equivalent to the generalized Pareto distribution with location , scale and shape and, conversely, one can get the Pareto distribution from the GPD by taking and if .
Bounded Pareto distribution
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 probability density function is
where L ≤ x ≤ H, and α > 0.
Generating bounded Pareto random variables If U is uniformly distributed on (0, 1), then applying inverse-transform method
is a bounded Pareto-distributed.
Symmetric Pareto distribution The purpose of the Symmetric and Zero Symmetric Pareto distributions is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from the Pareto distribution. Long probability tails normally means that probability decays slowly, and can be used to fit a variety of datasets. 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.
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:
The corresponding probability density function (PDF) is:
This distribution has two parameters: a and b. It is symmetric about 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:
The corresponding PDF is:
This distribution is symmetric about zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.
Multivariate Pareto distribution The univariate Pareto distribution has been extended to a multivariate Pareto distribution.
Statistical inference
Estimation of parameters The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
Therefore, the logarithmic likelihood function is
It can be seen that is monotonically increasing with xm, that is, the greater the value of xm, the greater the value of the likelihood function. Hence, since x ≥ xm, we conclude that
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
Thus the maximum likelihood estimator for α is:
The expected statistical error is:
Malik (1970) gives the exact joint distribution of . In particular, and are independent and is Pareto with scale parameter xm and shape parameter nα, whereas has an inverse-gamma distribution with shape and scale parameters n − 1 and nα, respectively.
Occurrence and applications General