InverseGamma: Inverse Gamma Distribution Class In distr6: The Complete R6 Probability Distributions Interface

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.

Details

The Inverse Gamma distribution parameterised with shape, α, and scale, β, is defined by the pdf,

f(x) = (β^α)/Γ(α)x^{-α-1}exp(-β/x)

for α, β > 0, where Γ is the gamma function.

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)

N/A

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

Inherited methods

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) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

Usage
`InverseGamma\$mean(...)`
`...`

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

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
`InverseGamma\$variance(...)`
`...`

Unused.

Method `skewness()`

The skewness of a distribution is defined by the third standardised moment,

sk_X = E_X[((x - μ)/σ)^3]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution.

Usage
`InverseGamma\$skewness(...)`
`...`

Unused.

Method `kurtosis()`

The kurtosis of a distribution is defined by the fourth standardised moment,

k_X = E_X[((x - μ)/σ)^4]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ 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

- ∑ (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

mgf_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

pgf_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.

References

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

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`