WeibullMinFactory

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

WeibullMin factory.

See also

DistributionFactory, WeibullMin

Notes

Several estimators to build a WeibullMin distribution from a scalar sample are proposed.

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

The estimator (\hat{\beta}_n, \hat{\alpha}_n, \hat{\gamma}_n) of (\beta, \alpha, \gamma) is defined by :

(1)\begin{eqnarray*} \displaystyle \Hat{\gamma}_n & = & (1-\mathrm{sign}(x_{(1,n)})/(2+n))x_{(1,n)} \end{eqnarray*} \begin{eqnarray*} \overline{x}_n & = & \hat{\beta}_n \,\Gamma\left(1 + \frac{1}{\hat{\alpha}_n}\right) + \hat{\gamma}_n \\ s_n^2 & = & \hat{\beta}_n^2 \left( \Gamma \left( 1 + \frac{2}{\hat{\alpha}_n} \right) - \Gamma^2 \left( 1 + \frac{1}{\hat{\alpha}_n} \right) \right) \end{eqnarray*}

Maximum likelihood based estimator:

The following sums are defined by:

\begin{eqnarray*} S_0 &=& \sum_{i=1}^n \frac{1}{x_i - \gamma} \\ S_1 &=& \sum_{i=1}^n \log (x_i - \gamma) \\ S_2 &=& \sum_{i=1}^n (x_i - \gamma)^{\alpha} \log (x_i - \gamma) \\ S_3 &=& \sum_{i=1}^n (x_i - \gamma)^{\alpha}\\ S_4 &=& \sum_{i=1}^n (x_i - \gamma)^{\alpha-1} \end{eqnarray*}

The Maximum Likelihood estimator of (\beta, \alpha, \gamma) is defined by (\hat{\beta}_n, \hat{\alpha}_n, \hat{\gamma}_n) verifying:

(2)\begin{eqnarray*} S_3(\hat{\alpha}_n,\hat{\gamma}_n) - n\hat{\beta}_n^{\hat{\alpha}_n} = 0 \\ \hat{\alpha}_n \left[S_0(\hat{\gamma}_n) - n\dfrac{S_4(\hat{\alpha}_n,\hat{\gamma}_n)}{S_3(\hat{\alpha}_n,\hat{\gamma}_n)} \right] - S_0(\hat{\gamma}_n) = 0 \\ S_0(\hat{\gamma}_n)(S_3(\hat{\alpha}_n,\hat{\gamma}_n)(n+S_1(\hat{\gamma}_n))-nS_2(\hat{\alpha}_n,\hat{\gamma}_n))-n^2S_4(\hat{\alpha}_n,\hat{\gamma}_n) = 0 \end{eqnarray*}

Methods

build(*args)

Estimate the distribution using the default strategy.

buildAsWeibullMin(*args)

Estimate the distribution as native distribution.

buildEstimator(*args)

Build the distribution and the parameter distribution.

buildMethodOfLikelihoodMaximization(sample)

Method of likelihood maximization.

buildMethodOfMoments(sample)

Method of 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.

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.

setShadowedId(id)

Accessor to the object's shadowed id.

setVisibility(visible)

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

Estimate the distribution using the default strategy.

Available usages:

build(sample)

build(param)

Parameters
sampleSample

Data

paramCollection of PointWithDescription

A vector of parameters of the distribution.

Returns
distributionDistribution

The estimated distribution

Notes

The default strategy is using the maximum likelihood estimators.

buildAsWeibullMin(*args)

Estimate the distribution as native distribution.

Parameters
sampleSample

Data

Returns
distributionWeibullMin

The estimated 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.

buildMethodOfLikelihoodMaximization(sample)

Method of likelihood maximization.

Refer to MaximumLikelihoodFactory.

Parameters
sampleSample

Data

Returns
distributionWeibullMin

The estimated distribution

Notes

The maximisation of the likelihood is initialized with the value of the estimator calculated with the method of moments.

buildMethodOfMoments(sample)

Method of moments estimator.

Parameters
sampleSample

Data

Returns
distributionWeibullMin

The estimated distribution

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.

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.

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.