Mathematical and statistical functions for the Pareto distribution, which is commonly used in Economics to model the distribution of wealth and the 80-20 rule.

## Value

Returns an R6 object inheriting from class SDistribution.

## Details

The Pareto distribution parameterised with shape, $$\alpha$$, and scale, $$\beta$$, is defined by the pdf, $$f(x) = (\alpha\beta^\alpha)/(x^{\alpha+1})$$ for $$\alpha, \beta > 0$$.

Currently this is implemented as the Type I Pareto distribution, other types will be added in the future. Characteristic function is omitted as no suitable incomplete gamma function with complex inputs implementation could be found.

## Distribution support

The distribution is supported on $$[\beta, \infty)$$.

## Default Parameterisation

Pare(shape = 1, scale = 1)

N/A

N/A

## 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, InverseGamma, Laplace, Logistic, Loglogistic, Lognormal, MultivariateNormal, Normal, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull

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

## Super classes

distr6::Distribution -> distr6::SDistribution -> Pareto

## Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

alias

Alias of the distribution.

packages

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

## Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

## Methods

Inherited methods

### Method new()

Creates a new instance of this R6 class.

#### 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).

### 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.

#### 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.

#### 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.

#### 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

Pareto\$clone(deep = FALSE)

#### Arguments

deep

Whether to make a deep clone.