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

 Arcsine R Documentation

## Arcsine Distribution Class

### Description

Mathematical and statistical functions for the Arcsine distribution, which is commonly used in the study of random walks and as a special case of the Beta distribution.

### Details

The Arcsine distribution parameterised with lower, a, and upper, b, limits is defined by the pdf,

f(x) = 1/(π√{(x-a)(b-x))}

for -∞ < a ≤ b < ∞.

### Value

Returns an R6 object inheriting from class SDistribution.

### Distribution support

The distribution is supported on [a, b].

### Default Parameterisation

Arc(lower = 0, upper = 1)

N/A

N/A

### Super classes

distr6::Distribution -> distr6::SDistribution -> Arcsine

### Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

### Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

### Methods

#### Public methods

Inherited methods

#### Method new()

Creates a new instance of this R6 class.

##### Usage
Arcsine\$new(lower = NULL, upper = NULL, decorators = NULL)
##### Arguments
lower

(numeric(1))
Lower limit of the Distribution, defined on the Reals.

upper

(numeric(1))
Upper limit of the Distribution, defined on the 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
Arcsine\$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
Arcsine\$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
Arcsine\$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
Arcsine\$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
Arcsine\$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
Arcsine\$entropy(base = 2, ...)
##### Arguments
base

(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)

...

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
Arcsine\$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
Arcsine\$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: BetaNoncentral, Beta, Cauchy, 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: 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, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

distr6 documentation built on March 28, 2022, 1:05 a.m.