Pareto distribution - Alchetron, The Free Social Encyclopedia



Parameters xm > 0 scale (real)α > 0 shape (real) Support x ∈ [ x m , + ∞ ) {displaystyle xin [x_{mathrm {m} },+infty )} PDF α x m α x α + 1 for x ≥ x m {displaystyle {rac {alpha ,x_{mathrm {m} }^{alpha }}{x^{alpha +1}}}{ ext{ for }}xgeq x_{m}} CDF 1 − ( x m x ) α for x ≥ x m {displaystyle 1-left({rac {x_{mathrm {m} }}{x}}ight)^{alpha }{ ext{ for }}xgeq x_{m}} Mean { ∞ for α ≤ 1 α x m α − 1 for α > 1 {displaystyle {egin{cases}infty &{ ext{for }}alpha leq 1{rac {alpha ,x_{mathrm {m} }}{alpha -1}}&{ ext{for }}alpha >1end{cases}}} Median x m 2 α {displaystyle x_{mathrm {m} }{sqrt[{alpha }]{2}}}

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 (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

F ¯ ( x ) = Pr ( X > x ) = { ( x m x ) α x ≥ x m , 1 x < x m ,

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.

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 ) = { 1 − ( x m x ) α x ≥ x m , 0 x < x m .

Probability density function

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

f X ( x ) = { α x m α x α + 1 x ≥ x m , 0 x < x m .

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.

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 x 1 exceeding x m , is a Pareto distribution with the same Pareto index α but with minimum x 1 instead of x m .

A characterization theorem

Suppose X 1 , X 2 , X 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 { X 1 , … , X n } and ( X 1 + ⋯ + X n ) / min { X 1 , … , X n } are independent. Then the common distribution is a Pareto distribution.

Geometric mean

The geometric mean (G) is

G = x m exp ⁡ ( 1 α ) .

Harmonic mean

The harmonic mean (H) is

H = x m ( 1 + 1 α ) .

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 (complementary CDF).

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

In this section, the symbol x_m, 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–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 ) = y γ 1 − 1 ( 1 − y ) γ 2 − 1 B ( γ 1 , γ 2 ) , 0 < y < 1 ; γ 1 , γ 2 > 0 ,

where B( ) is the beta function. If

W = μ + σ ( Y − 1 − 1 ) γ , σ > 0 , γ > 0 ,

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

If U 1 ∼ Γ ( δ 1 , 1 ) and U 2 ∼ Γ ( δ 2 , 1 ) are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is

W = μ + σ ( U 1 U 2 ) γ

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

F P ( σ , σ , 1 , 1 , α ) = P ( I ) ( σ , α ) F P ( μ , σ , 1 , 1 , α ) = P ( I I ) ( μ , σ , α ) F P ( μ , σ , γ , 1 , 1 ) = P ( I I I ) ( μ , σ , γ ) F P ( μ , σ , γ , 1 , α ) = P ( I V ) ( μ , σ , γ , α ) .

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. (Note that the Pareto distribution is not realistic for wealth for the lower end. 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:

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

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.

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 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 ⁡ ( X x m )

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

x 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 ( log ⁡ ( X x m ) < y ) = Pr ( X < x m e y ) = 1 − ( x m x m e y ) α = 1 − e − α y .

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 (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 x m and shape α is equivalent to the generalized Pareto distribution with location μ = x m , scale σ = x m / α and shape ξ = 1 / α . Vice versa one can get the Pareto distribution from the GPD by x m = σ / ξ and α = 1 / ξ .

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 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

L ( F ) = ∫ x m x ( F ) x f ( x ) d x ∫ x m ∞ x f ( x ) d x = ∫ 0 F x ( F ′ ) d F ′ ∫ 0 1 x ( F ′ ) d F ′

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

x ( F ) = x m ( 1 − F ) 1 α

and the Lorenz curve is calculated to be

L ( F ) = 1 − ( 1 − F ) 1 − 1 α ,

For 0 < α ≤ 1 the denominator is infinite, yielding L=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

According to Oxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population. We can estimate the Pareto index that would apply to this situation. Letting ε equal 62 / ( 7 × 10 9 ) we have:

L ( 1 / 2 ) = 1 − L ( 1 − ϵ )

or

1 − ( 1 / 2 ) 1 − 1 α = ϵ 1 − 1 α

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.

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 (for α ≥ 1 ) to be

G = 1 − 2 ( ∫ 0 1 L ( F ) d F ) = 1 2 α − 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 ( α , x m ) = ∏ i = 1 n α x m α x i α + 1 = α n x m n α ∏ i = 1 n 1 x i α + 1 .

Therefore, the logarithmic likelihood function is

ℓ ( α , x m ) = n ln ⁡ α + n α ln ⁡ x m − ( α + 1 ) ∑ i = 1 n ln ⁡ x i .

It can be seen that ℓ ( α , x m ) is monotonically increasing with x_m, that is, the greater the value of x_m, the greater the value of the likelihood function. Hence, since x ≥ x_m, we conclude that

x ^ m = min i x i .

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

∂ ℓ ∂ α = n α + n ln ⁡ x m − ∑ i = 1 n ln ⁡ x i = 0.

Thus the maximum likelihood estimator for α is:

α ^ = n ∑ i ( ln ⁡ x i − ln ⁡ x ^ m ) .

The expected statistical error is:

σ = α ^ n .

Malik (1970) gives the exact joint distribution of ( x ^ m , α ^ ) . In particular, x ^ m and α ^ are independent and x ^ m is Pareto with scale parameter x_m and shape parameter nα, whereas α ^ 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 ≥ x_m,

log ⁡ f X ( x ) = log ⁡ ( α x m α x α + 1 ) = log ⁡ ( α x m α ) − ( α + 1 ) log ⁡ x .

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

T = x m U 1 α

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

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 variance in the table on the right should be interpreted as the second moment).

The probability density function is

α L α x − α − 1 1 − ( L H ) α

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

U = 1 − L α x − α 1 − ( L H ) α x = ( − U H α − U L α − H α H α L α ) − 1 α

is a bounded Pareto-distributed.

Symmetric Pareto distribution

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

f ( x ; α , x m ) = { 1 2 α x m α | x | − α − 1 | x | > x m 0 otherwise .

It has a similar shape to a Pareto distribution for x > x_m and is mirror symmetric about the vertical axis.

Multivariate Pareto distribution

The univariate Pareto distribution has been extended a multivariate Pareto distribution.

References

Pareto distribution Wikipedia

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