Generalized Pareto Distribution (GPD)

Format: GPD(a, b, c)

Imagine that we are interested in general terms in the possible shape of the tails of a distribution. After all, that is where the big risks lie, so the more precisely we can determine this the better our estimate of the low probability, high impact events will be.

The distribution function  of the amount by which some variable X exceeds a threshold t given that it is above that threshold is called the conditional excess distribution function and is defined as:

Pickands (1975) showed that for a large class of underlying distribution functions the conditional excess distribution function for larger values of y follows what we now call the GPD distribution.
 

Because of its derivation by Pickands, the GPD is most commonly used in two situations:

  1. When spliced together with another continuous distribution, where the other distribution (e.g. a Lognormal) models the lower values of the variable, and the GPD models the high value tail; and

  2. When data are collected for some random variable only if the values exceed some large threshold. Then the GPD can be used to fit to those data.  

The three parameters defining the distribution are:

a - the minimum value the variable can take. Note that when splicing with another distribution, a will be the value at which the distributions are spliced

b - scale. Increasing the value b increases the spread (e.g. standard deviation)

c - shape. Greatly affects the distribution’s shape. If c<0 , the distribution has a maximum of (a-b)/c. If c=-1, the distribution is equivalent to a Uniform(a,b). If c<-1 the distribution is left-skewed, and if c>-1 it is right-skewed. Since the GPD is generally used to model the right tail of a random variable, in practice c is usually greater than zero.

The great benefit of the GPD is that it will take a continuous range of possible shapes that includes both the exponential and Pareto distributions as special cases. The generalized Pareto distribution allows you to "let the data decide" which distribution is appropriate, instead of having to select a particular form.

The Generalized Pareto distribution has three basic shapes, each corresponding to a limiting distribution of exceedence from a different class of underlying distributions:

  1. Distributions whose tails decrease exponentially, such as the normal, lead to a generalized Pareto distribution with shape parameter c=0
  2. Distributions whose tails decrease as a polynomial, such as Student's t, give c>0
  3. Distributions whose tails are finite, such as the PERT of Triangle, lead to  c<0

Other interesting things to know about the GPD

The GPD has applications in the analysis of extreme events, in the modelling of large insurance claims, as a failure time distribution and other situations where the Exponential distribution would seem appropriate but gives poor fit to data in its tail.

The instantaneous failure rate of the GPD is given by:

and is therefore monotonic in x, decreasing if c>0, increasing if, c<0 and constant (i.e. an Exponential distribution) if c=0.

If a random variable X follows a GPD, then for some threshold value t, the distribution of X-t  conditional on X>t also follows a GPD.

Relationship to other distributions

GPB(a,b,-1) = Uniform(a,b)

GPB(0,b,0) = Exponential(b)

GPD(0,b,c) = Pareto2(c/b, 1/c)

 If Y~Exponential(1) then    follows a GPD(0,b,c) distribution. 

ModelRisk functions added to Microsoft Excel for the Generalized Pareto distribution (GPD)

VoseGPD generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseGPDObject constructs a distribution object for this distribution.

VoseGPDProb returns the probability density or cumulative distribution function for this distribution.

VoseGPDProb10 returns the log10 of the probability density or cumulative distribution function.

VoseGPDFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseGPDFitObject constructs a distribution object of this distribution fitted to data.

VoseGPDFitP returns the parameters of this distribution fitted to data.

 

Generalized Pareto distribution equations

 

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