GeneralizedParetoDistribution

Generalized Pareto probability distribution object

Description

A GeneralizedParetoDistribution object consists of parameters, a model description, and sample data for a generalized Pareto probability distribution.

The generalized Pareto distribution is used to model the tails of another distribution. It allows a continuous range of possible shapes that include both the exponential and Pareto distributions as special cases. It has three basic forms, each corresponding to a limiting distribution of exceedance data from a different class of underlying distributions.

• Distributions whose tails decrease exponentially, such as the normal, lead to a generalized Pareto shape parameter of zero.

• Distributions whose tails decrease polynomially, such as the Student’s t, lead to a positive shape parameter.

• Distributions whose tails are finite, such as the beta, lead to a negative shape parameter.

The generalized Pareto distribution uses the following parameters.

Parameter	Description	Support
k	Shape parameter	$$-\infty <k<\infty$$
sigma	Scale parameter	$$\sigma \ge 0$$
theta	Location (threshold) parameter	$$-\infty <\theta <\infty$$

Creation

There are several ways to create a GeneralizedParetoDistribution probability distribution object.

• Create a distribution with specified parameter values using makedist.

• Fit a distribution to data using fitdist.

• Interactively fit a distribution to data using the Distribution Fitter app.

Properties

Distribution Parameters

k — Shape parameter

scalar value

Shape parameter for the generalized Pareto distribution, specified as a scalar value.

Data Types: single | double

sigma — Scale parameter

nonnegative scalar value

Scale parameter for the generalized Pareto distribution, specified as a nonnegative scalar value.

Data Types: single | double

theta — Location (threshold) parameter

scalar value

Location (threshold) parameter for the generalized Pareto distribution, specified as a scalar value.

Data Types: single | double

Distribution Characteristics

IsTruncated — Logical flag for truncated distribution

0 | 1

This property is read-only.

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

NumParameters — Number of parameters

positive integer value

This property is read-only.

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

ParameterCovariance — Covariance matrix of the parameter estimates

matrix of scalar values

This property is read-only.

Covariance matrix of the parameter estimates, specified as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0.

Data Types: double

ParameterIsFixed — Logical flag for fixed parameters

array of logical values

This property is read-only.

Logical flag for fixed parameters, specified as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed.

Data Types: logical

ParameterValues — Distribution parameter values

vector of scalar values

This property is read-only.

Distribution parameter values, specified as a vector of scalar values.

Data Types: single | double

Truncation — Truncation interval

vector of scalar values

This property is read-only.

Truncation interval for the probability distribution, specified as a vector of scalar values containing the lower and upper truncation boundaries.

Data Types: single | double

Other Object Properties

DistributionName — Probability distribution name

character vector

This property is read-only.

Probability distribution name, specified as a character vector.

Data Types: char

InputData — Data used for distribution fitting

structure

This property is read-only.

Data used for distribution fitting, specified as a structure containing the following:

• data: Data vector used for distribution fitting.

• cens: Censoring vector, or empty if none.

• freq: Frequency vector, or empty if none.

Data Types: struct

ParameterDescription — Distribution parameter descriptions

cell array of character vectors

This property is read-only.

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

ParameterNames — Distribution parameter names

cell array of character vectors

This property is read-only.

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

Object Functions

cdf	Cumulative distribution function
gather	Gather properties of Statistics and Machine Learning Toolbox object from GPU
icdf	Inverse cumulative distribution function
iqr	Interquartile range of probability distribution
mean	Mean of probability distribution
median	Median of probability distribution
negloglik	Negative loglikelihood of probability distribution
paramci	Confidence intervals for probability distribution parameters
pdf	Probability density function
plot	Plot probability distribution object
proflik	Profile likelihood function for probability distribution
random	Random numbers
std	Standard deviation of probability distribution
truncate	Truncate probability distribution object
var	Variance of probability distribution

Examples

Create a Generalized Pareto Distribution Object Using Default Parameters

Create a generalized Pareto distribution object using the default parameter values.

pd = makedist('GeneralizedPareto')

pd = 
  GeneralizedParetoDistribution

  Generalized Pareto distribution
        k = 1
    sigma = 1
    theta = 1

Create a Generalized Pareto Distribution Object Using Specified Parameters

Create a generalized Pareto distribution object by specifying parameter values.

pd = makedist('GeneralizedPareto','k',0,'sigma',2,'theta',1)

pd = 
  GeneralizedParetoDistribution

  Generalized Pareto distribution
        k = 0
    sigma = 2
    theta = 1

Compute the mean of the distribution.

m = mean(pd)

m = 
3

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• GeneralizedParetoDistribution can be a probability distribution object fitted by using fitdist with GPU array input arguments.

• The object functions of GeneralizedParetoDistribution fully support GPU arrays.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2013a