ParetoPickandsDistribution

ParetoPickandsDistribution[μ,σ,ξ]

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

ParetoPickandsDistribution[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 values in the generalized Pareto distribution is proportional to for and , and for for and . »
  • The survival function for values 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:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

Cumulant:

FactorialMoment:

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 ^(th) 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 , cumulative distributions functions are also 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:

Introduced in 2019
 (12.0)