Probability density function for the standard Pareto–Pickands distribution:

Hazard function for the standard Pareto–Pickands distribution:

Mean of the Pareto–Pickands distribution:

Standard deviation of the Pareto–Pickands distribution:

Median of the Pareto–Pickands distribution:

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 Pareto–Pickands distribution only depends on the shape parameter where defined:

Limiting values:

The minimal value of the kurtosis within the family of Pareto–Pickands distributions:

Table of moments of Pareto–Pickands distribution:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

Cumulant:

FactorialMoment:

Quantile function of Pareto–Pickands distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

Quartile deviation:

Model a tail of a power‐tail distribution, e.g. StudentTDistribution:

Truncate data to the left at :

Fit truncated data to Pareto–Pickands 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 Pareto–Pickands family of distributions:

Illustrate this fact with a sequence of probability plots of exceedance data against Pareto–Pickands family of distributions:

Pareto–Pickands distribution family is closed under affine transformations:

Pareto–Pickands distribution family is closed under left truncation (threshold stability):

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

Pareto–Pickands distribution with is equivalent to a UniformDistribution:

Pareto–Pickands distribution with is equivalent to a shifted ExponentialDistribution:

The Pareto–Pickands distribution family includes ParetoDistribution of types I and II:

Check that probability density functions coincide:

Standard Pareto–Pickands distribution with a positive shape parameter ξ is a special case of TsallisQExponentialDistribution:

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

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

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