The Pareto distribution, named after the Italian civil engineer, economist, and sociologist
Vilfredo Pareto, is a power law probability distribution that is used in description of
social, scientific, geophysical, actuarial, and many other types of observable phenomena.
Definition If X is a random variable with a Pareto distribution,
then the probability that X is greater than some number x, i.e. the survival function,
is given by
where xm is the 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.
Properties Cumulative distribution function
From the definition, the cumulative distribution function of a Pareto random variable with
parameters α and xm 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.
When plotted in a log-log plot, the distribution is represented by a straight line.
Probability density function It follows that the probability density function
is
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
The characteristic function is given by
where Γ(a, x) is the incomplete gamma function. 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 xm,
is a Pareto distribution with the same Pareto index α but with minimum x1 instead of xm.
A characterization theorem Suppose X1, X2, X3, ... are independent identically
distributed random variables whose probability distribution is supported on the interval
[xm, ∞) for some xm > 0. Suppose that for all n, the two random variables min{X1, ..., Xn}
and/min{X1, ..., Xn} are independent. Then the common distribution is a Pareto distribution.
Geometric mean The geometric mean is
Harmonic mean The harmonic mean is
Generalized Pareto distributions
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–III as special cases. The Feller–Pareto
distribution generalizes Pareto Type IV. Pareto types I–IV
The Pareto distribution hierarchy is summarized in the next table comparing the survival functions.
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. 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
are
The finiteness of the mean, and the existence and the finiteness of the variance depend
on the tail index α. 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–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 variable
is
and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution
are
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 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
File size distribution of Internet traffic which uses the TCP protocol
Hard disk drive error rates Clusters of Bose–Einstein condensate near
absolute zero The values of oil reserves in oil fields
The length distribution in jobs assigned supercomputers The standardized price returns on individual
stocks
Sizes of sand particles Sizes of meteorites
Numbers of species per genus 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 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. 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.
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, or bounded both above and below, 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 . Vice versa
one can get the Pareto distribution from the GPD by and .
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 n > 0, 100pn % of all people receive 100(1 − p)n
percentage 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
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
where x(F) is the inverse of the CDF. For the Pareto distribution,
and the Lorenz curve is calculated to be
Although the numerator and denominator in the expression for diverge for , their ratio
does not, yielding L=0 in these cases, which yields a Gini coefficient of unity. 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 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
(see Aaberge 2005). Parameter estimation
The likelihood function for the Pareto distribution parameters α and xm, given a 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 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.
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.
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
is Pareto-distributed. If U is uniformly distributed on [0, 1), it can be exchanged with.
Variants Bounded Pareto distribution
The bounded 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, then applying inverse-transform method
is a bounded Pareto-distributed. Symmetric Pareto distribution
The symmetric Pareto distribution can be defined by the probability density function:
It has a similar shape to a Pareto distribution for x > xm 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
Notes
References M. O. Lorenz. "Methods of measuring the concentration
of wealth". Publications of the American Statistical Association 9: 209–219. Bibcode:1905PAmSA...9..209L.
doi:10.2307/2276207. Pareto V "La Courbe de la Repartition de la
Richesse". In: Busino G, editor. Oevres Completes de Vilfredo Pareto. Geneva: Librairie Droz.
pp. 1–5. Pareto, V.. La legge della domanda. Giornale
degli Economisti, 10, 59–68. English translation in Rivista di Politica Economica, 87, 691–700.
Pareto, V.. Cours d'économie politique. Lausanne: Ed. Rouge.
External links Gini's Nuclear Family / Rolf Aabergé. – In:
International Conference to Honor Two Eminent Social Scientists, May, 2005 – PDF
Hazewinkel, Michiel, ed., "Pareto distribution", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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", MathWorld.