# ParetoPickandsDistribution

ParetoPickandsDistribution[μ,σ,ξ]

gives a ParetoPickands distribution with location parameter μ, scale parameter σ and shape parameter ξ.

gives the standard ParetoPickands distribution with zero location and unit scale parameters.

# Details

• The ParetoPickandsDistribution is also known as generalized Pareto distribution or GPD.
• ParetoPickandsDistribution allows σ to be any positive real number and μ and ξ to be any real numbers.
• The probability density function for value in the generalized Pareto distribution is proportional to for and , and for for and . »
• The survival function for value in the generalized Pareto distribution equals for and , and for for and .
• The hazard function for value in the generalized Pareto distribution equals for and , and zero otherwise. »
• ParetoPickandsDistribution can be used with such functions as Mean, CDF and RandomVariate.

# Examples

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## Basic Examples(4)

Probability density function for the standard ParetoPickands distribution:

Hazard function for the standard ParetoPickands distribution:

Mean of the ParetoPickands distribution:

Standard deviation of the ParetoPickands distribution:

Median of the ParetoPickands distribution:

## Scope(7)

Generate a sample of pseudorandom numbers from generalized Pareto distribution:

Compare data histogram to the population PDF:

Generate a sample of pseudorandom numbers from generalized Pareto distribution:

Estimate the distribution parameters from sample data:

Compare sample histogram with probability density functions of the estimated distribution:

Skewness of the generalized Pareto distribution depends only on ξ where defined:

Limiting values:

Invert skewness as a function of the shape parameter ξ:

Visualize the inverse function:

Kurtosis of ParetoPickands distribution only depends on the shape parameter where defined:

Limiting values:

The minimal value of the kurtosis within the family of ParetoPickands distributions:

Table of moments of ParetoPickands distribution:

Closed form for symbolic order:

Closed form for symbolic order:

Quantile function of ParetoPickands distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

Quartile deviation:

## Applications(2)

Model a tail of a powertail distribution, e.g. StudentTDistribution:

Truncate data to the left at :

Fit truncated data to ParetoPickands distribution:

Generate a sample from standard Gaussian distribution:

Define a function to extract largest elements of the sample while shifting the largest element to zero:

Exceedance in large data samples is well described by ParetoPickands family of distributions:

Illustrate this fact with a sequence of probability plots of exceedance data against ParetoPickands family of distributions:

## Properties & Relations(8)

ParetoPickands distribution family is closed under affine transformations:

ParetoPickands distribution family is closed under left truncation (threshold stability):

Refine the result, assuming that truncation point belongs to the support of ParetoPickandsDistribution:

ParetoPickands distribution with is equivalent to a UniformDistribution:

ParetoPickands distribution with is equivalent to a shifted ExponentialDistribution:

The ParetoPickands distribution family includes ParetoDistribution of types I and II:

Check that probability density functions coincide:

Standard ParetoPickands distribution with a positive shape parameter ξ is a special case of TsallisQExponentialDistribution:

Standard ParetoPickands distributions are stochastically ordered, i.e. for any two parameters , cumulative distributions functions are (reverse) ordered for all :

ParetoPickands distribution with positive shape parameter ξ occurs as a parametric mixture of ExponentialDistribution whose rate follows a GammaDistribution:

## Possible Issues(1)

An alternative parameterization with negated shape parameter can sometimes be encountered in the literature:

Wolfram Research (2019), ParetoPickandsDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ParetoPickandsDistribution.html.

#### Text

Wolfram Research (2019), ParetoPickandsDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ParetoPickandsDistribution.html.

#### CMS

Wolfram Language. 2019. "ParetoPickandsDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ParetoPickandsDistribution.html.

#### APA

Wolfram Language. (2019). ParetoPickandsDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ParetoPickandsDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_paretopickandsdistribution, author="Wolfram Research", title="{ParetoPickandsDistribution}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/ParetoPickandsDistribution.html}", note=[Accessed: 20-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_paretopickandsdistribution, organization={Wolfram Research}, title={ParetoPickandsDistribution}, year={2019}, url={https://reference.wolfram.com/language/ref/ParetoPickandsDistribution.html}, note=[Accessed: 20-June-2024 ]}