GeneralizedParetoFactory

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../../_images/openturns-GeneralizedParetoFactory-1.png
class GeneralizedParetoFactory(*args)

Generalized Pareto factory.

Available constructor:

GeneralizedParetoFactory()

See also

DistributionFactory, GeneralizedPareto

Notes

Several estimators are available to build a GeneralizedPareto distribution from a scalar sample (see [matthys2003] for the theory).

The chosen algorithm depends on the size of the sample compared to the ResourceMap key GeneralizedParetoFactory-SmallSize:

  • When the size of the sample is small the probability weighted method of moments is used, with a fallback on the exponential regression method

  • When the size of the sample is large the exponential regression method is used, with a fallback on the probability weighted moment method

Moments based estimator:

Lets denote:

  • \displaystyle \overline{x}_n = \frac{1}{n} \sum_{i=1}^n x_i the empirical mean of the sample,

  • \displaystyle s_n^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x}_n)^2 its empirical variance.

Then we estimate (\hat{\sigma}_n, \hat{\xi}_n, \hat{u}_n) using:

(1)\begin{eqnarray*} \hat{\xi}_n &= \dfrac{1}{2}\left(\dfrac{\overline{x}_n^2}{s_n^2}-1\right) \\ \hat{\sigma}_n &= \dfrac{\overline{x}_n}{2}\left(\dfrac{\overline{x}_n^2}{s_n^2}+1\right) \\ \hat{u}_n &= x_{(1)} - \frac{x_{(1)}}{2 + n} \end{eqnarray*}

This estimator is well-defined only if \hat{\xi}>-1/4, otherwise the second moment does not exist.

Probability weighted moments based estimator:

Lets denote:

  • \left(x_{(i)}\right)_{i\in\{1,\dots,n\}} the sample sorted in ascending order

  • m=\dfrac{1}{n}\sum_{i=1}^n\left(1-\dfrac{i-7/20}{n}\right)x_{(i)}

  • \rho=\dfrac{m}{\overline{x}_n}

Then we estimate (\hat{\sigma}, \hat{\xi}, \hat{u}) using:

(2)\begin{eqnarray*} \hat{\xi}_n &= \dfrac{1-4\rho}{1-2\rho} \\ \hat{\sigma}_n &= \dfrac{2\overline{x}_n}{1-2\rho} \\ \hat{u}_n &= x_{(1)} - \frac{x_{(1)}}{2 + n} \end{eqnarray*}

This estimator is well-defined only if \hat{\xi}_n>-1/2, otherwise the first moment does not exist.

Maximum likelihood based estimator:

These estimators are not yet implemented.

For a given u < x_{(1,n)}, we get (\hat{\sigma}_n(u), \hat{\xi}_n(u) by maximizing the likelihood of the sample \ell(\sigma(u), \xi(u),u):

(\hat{\sigma}_n(u), \hat{\xi}_n(u)) = \argmax_{\sigma, \xi} \ell(\sigma(u), \xi(u)| x_1, \dots, x_n, u)

The threshold u is obtained by maximizing the optimal likelihood \ell(\hat{\sigma}_n(u), \hat{\xi}_n(u), u)

\hat{u}_n = \argmax_{u} \ell(\hat{\sigma}_n(u), \hat{\xi}_n(u),u) \text{ under the constraint } u < x_{(1,n)}

The initial point of the optimisation problem is u_0 = x_{(1,n)} - |x_{(1,n)}|/(2+n).

Exponential regression based estimator:

Lets denote:

  • y_{i}=i\log\left(\dfrac{x_{(n-i)}-x_{(1)}}{x_{(n-i-1)}-x_{(1)}}\right) for i\in\{1,n-3\}

Then we estimate (\hat{\sigma}, \hat{\xi}, \hat{u}) using:

(3)\hat{\xi} &= \xi^* \\ \hat{\sigma} &= \dfrac{2\overline{x}_n}{1-2\rho} \\ \hat{u} &= x_{(1)} - \frac{x_{(1)}}{2 + n}

Where \xi^* maximizes:

(4)\sum_{i=1}^{n-2}\log\left(\dfrac{1-(j/n)^{\xi}}{\xi}\right)-\dfrac{1-(j/n)^{\xi}}{\xi}y_i

under the constraint -1 \leq \xi \leq 1.

Methods

build(*args)

Build the distribution.

buildAsGeneralizedPareto(*args)

Build the distribution as a GeneralizedPareto type.

buildEstimator(*args)

Build the distribution and the parameter distribution.

buildMethodOfExponentialRegression(sample)

Build the distribution based on the exponential regression estimator.

buildMethodOfMoments(sample)

Build the distribution based on the method of moments estimator.

buildMethodOfProbabilityWeightedMoments(sample)

Build the distribution based on the probability weighted moments estimator.

getBootstrapSize()

Accessor to the bootstrap size.

getClassName()

Accessor to the object's name.

getId()

Accessor to the object's id.

getName()

Accessor to the object's name.

getOptimizationAlgorithm()

Accessor to the solver.

getShadowedId()

Accessor to the object's shadowed id.

getVisibility()

Accessor to the object's visibility state.

hasName()

Test if the object is named.

hasVisibleName()

Test if the object has a distinguishable name.

setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

setName(name)

Accessor to the object's name.

setOptimizationAlgorithm(solver)

Accessor to the solver.

setShadowedId(id)

Accessor to the object's shadowed id.

setVisibility(visible)

Accessor to the object's visibility state.
__init__(*args)
build(*args)

Build the distribution.

Available usages:

build()

build(sample)

build(param)

Parameters
sample2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

paramCollection of PointWithDescription

A vector of parameters of the distribution.

Returns
distDistribution

The built distribution.

Notes

In the first usage, the default GeneralizedPareto distribution is built.

In the second usage, the parameters are evaluated according the following strategy:

  • If the sample size is less or equal to GeneralizedParetoFactory-SmallSize from ResourceMap, then the method of probability weighted moments is used. If it fails, the method of exponential regression is used.

  • Otherwise, the first method tried is the method of exponential regression, then the method of probability weighted moments if the first one fails.

In the third usage, a GeneralizedPareto distribution corresponding to the given parameters is built.

buildAsGeneralizedPareto(*args)

Build the distribution as a GeneralizedPareto type.

Available usages:

build()

build(sample)

build(param)

Parameters
sample2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

paramsequence of float

A vector of parameters of the distribution.

Returns
distGeneralizedPareto

The built distribution.

buildEstimator(*args)

Build the distribution and the parameter distribution.

Parameters
sample2-d sequence of float

Sample from which the distribution parameters are estimated.

parametersDistributionParameters

Optional, the parametrization.

Returns
resDistDistributionFactoryResult

The results.

Notes

According to the way the native parameters of the distribution are estimated, the parameters distribution differs:

  • Moments method: the asymptotic parameters distribution is normal and estimated by Bootstrap on the initial data;

  • Maximum likelihood method with a regular model: the asymptotic parameters distribution is normal and its covariance matrix is the inverse Fisher information matrix;

  • Other methods: the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting (see KernelSmoothing).

If another set of parameters is specified, the native parameters distribution is first estimated and the new distribution is determined from it:

  • if the native parameters distribution is normal and the transformation regular at the estimated parameters values: the asymptotic parameters distribution is normal and its covariance matrix determined from the inverse Fisher information matrix of the native parameters and the transformation;

  • in the other cases, the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting.

buildMethodOfExponentialRegression(sample)

Build the distribution based on the exponential regression estimator.

Parameters
sample2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

buildMethodOfMoments(sample)

Build the distribution based on the method of moments estimator.

Parameters
sample2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

buildMethodOfProbabilityWeightedMoments(sample)

Build the distribution based on the probability weighted moments estimator.

Parameters
sample2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

getBootstrapSize()

Accessor to the bootstrap size.

Returns
sizeinteger

Size of the bootstrap.

getClassName()

Accessor to the object’s name.

Returns
class_namestr

The object class name (object.__class__.__name__).

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOptimizationAlgorithm()

Accessor to the solver.

Returns
solverOptimizationAlgorithm

The solver used for numerical optimization of the likelihood.

getShadowedId()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

True if the name is not empty and not the default one.

setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

Parameters
sizeinteger

Size of the bootstrap.

setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setOptimizationAlgorithm(solver)

Accessor to the solver.

Parameters
solverOptimizationAlgorithm

The solver used for numerical optimization of the likelihood.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.