distr6 (version 1.6.4)

InverseGamma: Inverse Gamma Distribution Class

Description

Mathematical and statistical functions for the Inverse Gamma distribution, which is commonly used in Bayesian statistics as the posterior distribution from the unknown variance in a Normal distribution.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on the Positive Reals.

Default Parameterisation

InvGamma(shape = 1, scale = 1)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> InverseGamma

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

• InverseGamma$new()

• InverseGamma$mean()

• InverseGamma$mode()

• InverseGamma$variance()

• InverseGamma$skewness()

• InverseGamma$kurtosis()

• InverseGamma$entropy()

• InverseGamma$mgf()

• InverseGamma$pgf()

• InverseGamma$clone()

• distr6::Distribution$cdf()

• distr6::Distribution$correlation()

• distr6::Distribution$getParameterValue()

• distr6::Distribution$iqr()

• distr6::Distribution$liesInSupport()

• distr6::Distribution$liesInType()

• distr6::Distribution$median()

• distr6::Distribution$parameters()

• distr6::Distribution$pdf()

• distr6::Distribution$prec()

• distr6::Distribution$print()

• distr6::Distribution$quantile()

• distr6::Distribution$rand()

• distr6::Distribution$setParameterValue()

• distr6::Distribution$stdev()

• distr6::Distribution$strprint()

• distr6::Distribution$summary()

• distr6::Distribution$workingSupport()

Method new()

Creates a new instance of this R6 class.

Usage

InverseGamma$new(shape = NULL, scale = NULL, decorators = NULL)

Arguments

shape

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

scale

(numeric(1)) Scale parameter, defined on the positive 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 $$E_X(X)=\sum p_X(x)\ast x$$ with an integration analogue for continuous distributions.

Usage

InverseGamma$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

InverseGamma$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 variance()

The variance of a distribution is defined by the formula $$va{r}_X=E[X^2]-E[X{]}^2$$ where $E_X$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

Usage

InverseGamma$variance(...)

Arguments

...

Unused.

Method skewness()

The skewness of a distribution is defined by the third standardised moment, $$s{k}_X=E_X[{\frac{x-\mu }{\sigma }}^3]$$ 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

InverseGamma$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

InverseGamma$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

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

Arguments

base

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

...

Unused.

Method mgf()

The moment generating function is defined by $$mg{f}_X(t)=E_X[exp(xt)]$$ where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

InverseGamma$mgf(t, ...)

Arguments

t

(integer(1)) t integer to evaluate function at.

...

Unused.

Method pgf()

The probability generating function is defined by $$pg{f}_X(z)=E_X[exp(z^x)]$$ where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

InverseGamma$pgf(z, ...)

Arguments

z

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

...

Unused.

Method clone()

The objects of this class are cloneable with this method.

Usage

InverseGamma$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Details

The Inverse Gamma distribution parameterised with shape, $\alpha$, and scale, $\beta$, is defined by the pdf, $$f(x)=({\beta }^{\alpha })/\mathrm{\Gamma }(\alpha )x^{-\alpha -1}exp(-\beta /x)$$ for $\alpha ,\beta >0$, where $\mathrm{\Gamma }$ is the gamma function.

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, Frechet, Gamma, Gompertz, Gumbel, 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, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete