gpstat

Generalized Pareto mean and variance

Syntax

m = gpstat(k,sigma,theta)

[m,v] = gpstat(k,sigma,theta)

Description

m = gpstat(k,sigma,theta) returns the mean for the generalized Pareto (GP) distribution with the shape parameter k, scale parameter sigma, and (threshold) location parameter theta.

[m,v] = gpstat(k,sigma,theta) also returns the variance for the distribution.

Examples

Compute Mean and Variance

Compute the mean and variance of a generalized Pareto distribution with the shape parameter k, scale parameter sigma, and location parameter theta.

k = 0.4;
sigma = 2;
theta = 3;
[mean,variance] = gpstat(k,sigma,theta)

mean = 
6.3333

variance = 
55.5556

Input Arguments

k — Shape parameter

scalar value | array of scalar values

Shape parameter, specified as a scalar value or an array of scalar values. If one or more of the input arguments k, sigma, and theta are arrays, then the array sizes must be the same. In this case, gpstat expands each scalar input into a constant array of the same size as the array inputs.

Data Types: single | double

sigma — Scale parameter

positive scalar value | array of positive scalar values

Scale parameter, specified as a positive scalar value or an array of positive scalar values. If one or more of the input arguments k, sigma, and theta are arrays, then the array sizes must be the same. In this case, gpstat expands each scalar input into a constant array of the same size as the array inputs.

Data Types: single | double

theta — Location parameter

0 (default) | scalar value | array of scalar values

Location parameter, specified as a scalar value or an array of scalar values. If one or more of the input arguments k, sigma, and theta are arrays, then the array sizes must be the same. In this case, gpstat expands each scalar input into a constant array of the same size as the array inputs.

Data Types: single | double

Output Arguments

m — Mean

numeric scalar | array of numeric scalars

Mean for the GP distribution, returned as a numeric scalar or an array of numeric scalars. m is the same size as k, sigma, and theta. Each element in m is the mean of the GP distribution specified by the corresponding elements in k, sigma, and theta.

When k = 0 and theta = 0, the generalized Pareto (GP) distribution is equivalent to the exponential distribution. When k > 0 and theta = sigma/k, the GP distribution is equivalent to a Pareto distribution with a scale parameter equal to sigma/k and a shape parameter equal to 1/k. The mean of the GP distribution is not finite when k ≥ 1, and the variance is not finite when k ≥ 1/2.

v — Variance

numeric scalar | array of numeric scalars

Variance for the GP distribution, returned as a numeric scalar or an array of numeric scalars. v is the same size as k, sigma, and theta. Each element in v is the variance of the GP distribution specified by the corresponding elements in k, sigma, and theta.

References

[1] Embrechts, P., C. Klüppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. New York: Springer, 1997.

[2] Kotz, S., and S. Nadarajah. Extreme Value Distributions: Theory and Applications. London: Imperial College Press, 2000.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

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

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a