# Empirical: Empirical Distribution Class In distr6: The Complete R6 Probability Distributions Interface

## Description

Mathematical and statistical functions for the Empirical distribution, which is commonly used in sampling such as MCMC.

## Details

The Empirical distribution is defined by the pmf,

p(x) = ∑ I(x = x_i) / k

for x_i ε R, i = 1,...,k.

Sampling from this distribution is performed with the sample function with the elements given as the support set and uniform probabilities. Sampling is performed with replacement, which is consistent with other distributions but non-standard for Empirical distributions. Use simulateEmpiricalDistribution to sample without replacement.

The cdf and quantile assumes that the elements are supplied in an indexed order (otherwise the results are meaningless).

## Value

Returns an R6 object inheriting from class SDistribution.

## Distribution support

The distribution is supported on x_1,...,x_k.

Emp(samples = 1)

N/A

N/A

## Super classes

distr6::Distribution -> distr6::SDistribution -> Empirical

## Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

## Methods

#### Public methods

Inherited methods

#### Method new()

Creates a new instance of this R6 class.

##### Usage
Empirical\$new(samples = NULL, decorators = NULL)
##### Arguments
samples

(numeric())
Vector of observed samples, see examples.

decorators

(character())
Decorators to add to the distribution during construction.

##### Examples
Empirical\$new(runif(1000))

#### 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
Empirical\$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
Empirical\$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
Empirical\$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
Empirical\$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
Empirical\$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
Empirical\$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
Empirical\$mgf(t, ...)
##### Arguments
t

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

...

Unused.

#### Method cf()

The characteristic function is defined by

cf_X(t) = E_X[exp(xti)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
Empirical\$cf(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
Empirical\$pgf(z, ...)
##### Arguments
z

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

...

Unused.

#### Method setParameterValue()

Sets the value(s) of the given parameter(s).

##### Usage
Empirical\$setParameterValue(
...,
lst = NULL,
error = "warn",
resolveConflicts = FALSE
)
##### Arguments
...

ANY
Named arguments of parameters to set values for. See examples.

lst

(list(1))
Alternative argument for passing parameters. List names should be parameter names and list values are the new values to set.

error

(character(1))
If "warn" then returns a warning on error, otherwise breaks if "stop".

resolveConflicts

(logical(1))
If FALSE (default) throws error if conflicting parameterisations are provided, otherwise automatically resolves them by removing all conflicting parameters.

#### Method clone()

The objects of this class are cloneable with this method.

##### Usage
Empirical\$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.