Generalized Pareto Distribution
These are the density and random generation functions for the generalized Pareto distribution.
This is a vector of data.
This is a positive scalar integer, and is the number of observations to generate randomly.
This is a scalar or vector location parameter mu. When xi is non-negative, mu must not be greater than x. When xi is negative, mu must be less than x + sigma/xi.
This is a positive-only scalar or vector of scale parameters sigma.
This is a scalar or vector of shape parameters xi.
Application: Continuous Univariate
Density: 1/sigma (1 + xi z)^(-1/xi + 1) where z = (theta - mu)/sigma
Inventor: Pickands (1975)
Notation 1: theta ~ GPD(mu, sigma, xi)
Notation 2: p(theta) ~ GPD(theta | mu, sigma, xi)
Parameter 1: location mu, where mu <= theta when xi >= 0, and mu >= theta + sigma/xi when xi < 0
Parameter 2: scale sigma > 0
Parameter 3: shape xi
Mean: mu + sigma / (1 - xi) when xi < 1
Variance: sigma^2 / [(1 - xi)^2 (1 - 2 xi)] when xi < 0.5
The generalized Pareto distribution (GPD) is a more flexible extension
of the Pareto (
dpareto) distribution. It is equivalent to
the exponential distribution when both mu = 0 and
xi = 0, and it is equivalent to the Pareto
distribution when mu = sigma / xi and
xi > 0.
The GPD is often used to model the tails of another distribution, and the shape parameter xi relates to tail-behavior. Distributions with tails that decrease exponentially are modeled with shape xi = 0. Distributions with tails that decrease as a polynomial are modeled with a positive shape parameter. Distributions with finite tails are modeled with a negative shape parameter.
dgpd gives the density, and
rgpd generates random deviates.
Pickands J. (1975). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics, 3, p. 119–131.
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