distr6 (version 1.6.0)

Cauchy: Cauchy Distribution Class

Description

Mathematical and statistical functions for the Cauchy distribution, which is commonly used in physics and finance.

Value

Returns an R6 object inheriting from class SDistribution.

Distribution support

The distribution is supported on the Reals.

Default Parameterisation

Cauchy(location = 0, scale = 1)

Omitted Methods

N/A

Also known as

N/A

Super classes

distr6::Distribution -> distr6::SDistribution -> Cauchy

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

• Cauchy$new()

• Cauchy$mean()

• Cauchy$mode()

• Cauchy$variance()

• Cauchy$skewness()

• Cauchy$kurtosis()

• Cauchy$entropy()

• Cauchy$mgf()

• Cauchy$cf()

• Cauchy$pgf()

• Cauchy$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

Cauchy$new(location = NULL, scale = NULL, decorators = NULL)

Arguments

location

(numeric(1)) Location parameter defined on the 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

Cauchy$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

Cauchy$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

Cauchy$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

Cauchy$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

Cauchy$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

Cauchy$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

Cauchy$mgf(t, ...)

Arguments

t

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

...

Unused.

Method cf()

The characteristic function is defined by $$c{f}_X(t)=E_X[exp(xti)]$$ where X is the distribution and $E_X$ is the expectation of the distribution X.

Usage

Cauchy$cf(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

Cauchy$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

Cauchy$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Details

The Cauchy distribution parameterised with location, $\alpha$, and scale, $\beta$, is defined by the pdf, $$f(x)=1/(\pi \beta (1+((x-\alpha )/\beta {)}^2))$$ for $\alpha ϵR$ and $\beta >0$.

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, ChiSquaredNoncentral, ChiSquared, Dirichlet, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, 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, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete