# Wolfram Mathematica

## Fitting Power-Tail Data with Pareto–Pickands Distribution

The ParetoPickands distribution, also known as generalized Pareto distribution (GPD), was introduced to study extreme order statistics and is often used to describe events exceeding some high threshold. The ParetoPickands distribution finds applications in insurance, hydrology and climatology.

The ParetoPickands distribution can also be used to model exceedance for powertail distributions, many of which exist in the Wolfram Language, like CauchyDistribution, DagumDistribution, LogGammaDistribution, ParetoDistribution, StudentTDistribution and more.

The higher moments of this distribution do not exist.

The histogram shows that the sample has a long tail.

Truncate data by keeping large values from the tail.

Fit the tail data to a ParetoPickands distribution.

Check goodness of fit with a quantile plot.

Goodness-of-fit test.

An interesting factoid: standard ParetoPickands distributions are stochastically ordered; i.e. for , cumulative distribution functions are also ordered for all :

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