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.
Returns an R6 object inheriting from class SDistribution.
The distribution is supported on \([\beta, \infty)\).
Pare(shape = 1, scale = 1)
N/A
N/A
distr6::Distribution -> distr6::SDistribution -> Pareto
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
packagesPackages required to be installed in order to construct the distribution.
propertiesReturns distribution properties, including skewness type and symmetry.
new()Creates a new instance of this R6 class.
Pareto$new(shape = NULL, scale = NULL, decorators = NULL)
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.
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.
Pareto$mean(...)
...Unused.
mode()The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).
Pareto$mode(which = "all")
which(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies
which mode to return.
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).
Pareto$median()
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.
Pareto$variance(...)
...Unused.
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.
Pareto$skewness(...)
...Unused.
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.
Pareto$kurtosis(excess = TRUE, ...)
excess(logical(1))
If TRUE (default) excess kurtosis returned.
...Unused.
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.
Pareto$entropy(base = 2, ...)
base(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)
...Unused.
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.
Pareto$mgf(t, ...)
t(integer(1))
t integer to evaluate function at.
...Unused.
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.
Pareto$pgf(z, ...)
z(integer(1))
z integer to evaluate probability generating function at.
...Unused.
clone()The objects of this class are cloneable with this method.
Pareto$clone(deep = FALSE)
deepWhether to make a deep clone.
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.
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,
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