Pareto distribution of the first kind




Format: Pareto(q, a)

 

The Pareto distributions have a ski slope shape: right skewed where mode and minimum are equal. The Pareto distribution of the first kind starts at the minimum value a, and has a rate of decrease determined by q: The larger the value of 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 over the entire claim size distribution.

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. There are other distributions with 'heavy tails' that are also used for modeling extremes.

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.

ModelRisk functions added to Microsoft Excel for the Pareto distribution of the First Kind

VosePareto generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseParetoObject constructs a distribution object for this distribution.

VoseParetoProb returns the probability density or cumulative distribution function for this distribution.

VoseParetoProb10 returns the log10 of the probability density or cumulative distribution function. 

VoseParetoFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseParetoFitObject constructs a distribution object of this distribution fitted to data.

VoseParetoFitP returns the parameters of this distribution fitted to data.

 

Pareto distribution of the first kind equations

 

 

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