Pareto(a,q)

Crystal Ball parameter restrictions

The Pareto distribution has an exponential type of shape: right skewed where mode and minimum are equal. It starts at a, and has a rate of decrease determined by q: The larger q, the quicker its tail falls away. Examples of the Pareto distribution are given below:

#### Uses

##### 1. Demographics

The Pareto distribution was originally used to model the number of people with an income of at least *x*, but it is now used to model any variable that has a minimum, but also most likely, value and for which the probability density decreases geometrically towards zero.

The Pareto distribution has also been used for city population sizes, occurrences of natural resources, stock price fluctuations, size of companies, personal income and error clustering in communication circuits.

An obvious use of the Pareto is for insurance claims. Insurance policies are written so that it is not worth claiming below a certain value (*a*) and the probability of a claim greater than *a* is assumed to decrease as a power function of the claim size. It turns out, however, that the Pareto distribution is generally a poor fit.

##### 2. Long-tailed variable

The Pareto distribution has the longest tail of all probability distributions. Thus, while it is not a good fit for the bulk of a variable like a claim size distribution, it is frequently used to model the tails by splicing with another distribution like a Lognormal. That way an insurance company is reasonably guaranteed to have a fairly conservative interpretation of what the (obviously rarely seen, but potentially catastrophic) very high claim values might be. It can also be used to model a longer-tailed discrete variable than any other distribution.

#### Comments

Named after Vilfredo Pareto (1848-1923), an Italian economist and sociologist: Pareto's law said that the distribution of incomes for a society follows a Pareto distribution. He believed it to be a universal law, but ... he was wrong. A lot of interest in income modeling is placed on the value of *q*: In the 19th century, this sat at around 1.7, and in the late 20th century it was about 2.0 in the developed world (Cramer, 1971). The Pareto tends to fit the tails of income distributions very well, but is a poor fit over the whole range, whereas the Lognormal distribution seems to fit well in the body of the distribution but poorly at the tails. In this example, we explain how you can combine (splice) different distributions such as the Gamma and Pareto distribution to effectively stretch the tail of a distribution you wish to use. The Pareto distribution is sometimes known as the *Zeta* distribution.

If a variable X = Pareto(a,*q*) then the variable Y = 1/X takes a Power Function distribution Power(a,*q*) with density f(x) = *q* a^{q}·y^{q-1}, 0 ≤ Y ≤ 1/a.