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distr6 (version 1.5.6)

Frechet: Frechet Distribution Class

Description

Mathematical and statistical functions for the Frechet distribution, which is commonly used as a special case of the Generalised Extreme Value distribution.

Arguments

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on \(x > \gamma\).

Default Parameterisation

Frec(shape = 1, scale = 1, minimum = 0)

Omitted Methods

N/A

Also known as

Also known as the Inverse Weibull distribution.

Super classes

distr6::Distribution -> distr6::SDistribution -> Frechet

Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

packages

Packages required to be installed in order to construct the distribution.

Methods

Public methods

Method new()

Creates a new instance of this R6 class.

Usage

Frechet$new(shape = NULL, scale = NULL, minimum = NULL, decorators = NULL)

Arguments

shape

(numeric(1)) Shape parameter, defined on the positive Reals.

scale

(numeric(1)) Scale parameter, defined on the positive Reals.

minimum

(numeric(1)) Minimum of the distribution, defined on the Reals.

decorators

(character()) Decorators to add to the distribution during construction.

Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions.

Usage

Frechet$mean(...)

Arguments

...

Unused.

Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

Usage

Frechet$mode(which = "all")

Arguments

which

(character(1) | numeric(1) Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.

Method median()

Returns the median of the distribution. If an analytical expression is available returns distribution median, otherwise if symmetric returns self$mean, otherwise returns self$quantile(0.5).

Usage

Frechet$median()

Method variance()

The variance of a distribution is defined by the formula $$var_X = E[X^2] - E[X]^2$$ where \(E_X\) is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

Frechet$variance(...)

Arguments

...

Unused.

Method skewness()

The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution.

Usage

Frechet$skewness(...)

Arguments

...

Unused.

Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Frechet$kurtosis(excess = TRUE, ...)

Arguments

excess

(logical(1)) If TRUE (default) excess kurtosis returned.

...

Unused.

Method entropy()

The entropy of a (discrete) distribution is defined by $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Frechet$entropy(base = 2, ...)

Arguments

base

(integer(1)) Base of the entropy logarithm, default = 2 (Shannon entropy)

...

Unused.

Method pgf()

The probability generating function is defined by $$pgf_X(z) = E_X[exp(z^x)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.

Usage

Frechet$pgf(z, ...)

Arguments

z

(integer(1)) z integer to evaluate probability generating function at.

...

Unused.

Method setParameterValue()

Sets the value(s) of the given parameter(s).

Usage

Frechet$setParameterValue(
  ...,
  lst = NULL,
  error = "warn",
  resolveConflicts = FALSE
)

Arguments

...

ANY Named arguments of parameters to set values for. See examples.

lst

(list(1)) Alternative argument for passing parameters. List names should be parameter names and list values are the new values to set.

error

(character(1)) If "warn" then returns a warning on error, otherwise breaks if "stop".

resolveConflicts

(logical(1)) If FALSE (default) throws error if conflicting parameterisations are provided, otherwise automatically resolves them by removing all conflicting parameters.

Method clone()

The objects of this class are cloneable with this method.

Usage

Frechet$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Details

The Frechet distribution parameterised with shape, \(\alpha\), scale, \(\beta\), and minimum, \(\gamma\), is defined by the pdf, $$f(x) = (\alpha/\beta)((x-\gamma)/\beta)^{-1-\alpha}exp(-(x-\gamma)/\beta)^{-\alpha}$$ for \(\alpha, \beta \epsilon R^+\) and \(\gamma \epsilon R\).

References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

See Also

Other continuous distributions: Arcsine, BetaNoncentral, Beta, Cauchy, ChiSquaredNoncentral, ChiSquared, Dirichlet, Erlang, Exponential, FDistributionNoncentral, FDistribution, Gamma, Gompertz, Gumbel, InverseGamma, Laplace, Logistic, Loglogistic, Lognormal, MultivariateNormal, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull

Other univariate distributions: Arcsine, Bernoulli, BetaNoncentral, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete