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

 Wald R Documentation

## Wald Distribution Class

### Description

Mathematical and statistical functions for the Wald distribution, which is commonly used for modelling the first passage time for Brownian motion.

### Details

The Wald distribution parameterised with mean, μ, and shape, λ, is defined by the pdf,

f(x) = (λ/(2x^3π))^{1/2} exp((-λ(x-μ)^2)/(2μ^2x))

for λ > 0 and μ > 0.

Sampling is performed as per Michael, Schucany, Haas (1976).

### Value

Returns an R6 object inheriting from class SDistribution.

### Distribution support

The distribution is supported on the Positive Reals.

### Default Parameterisation

Wald(mean = 1, shape = 1)

### Omitted Methods

`quantile` is omitted as no closed form analytic expression could be found, decorate with `FunctionImputation` for a numerical imputation.

### Also known as

Also known as the Inverse Normal distribution.

### Super classes

`distr6::Distribution` -> `distr6::SDistribution` -> `Wald`

### 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

Inherited methods

#### Method `new()`

Creates a new instance of this R6 class.

##### Usage
`Wald\$new(mean = NULL, shape = NULL, decorators = NULL)`
##### Arguments
`mean`

`(numeric(1))`
Mean of the distribution, location parameter, defined on the positive Reals.

`shape`

`(numeric(1))`
Shape 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) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

##### Usage
`Wald\$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
`Wald\$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
`Wald\$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
`Wald\$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
`Wald\$kurtosis(excess = TRUE, ...)`
##### Arguments
`excess`

`(logical(1))`
If `TRUE` (default) excess kurtosis returned.

`...`

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
`Wald\$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
`Wald\$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
`Wald\$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
`Wald\$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.

Michael, J. R., Schucany, W. R., & Haas, R. W. (1976). Generating random variates using transformations with multiple roots. The American Statistician, 30(2), 88-90.

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`, `Pareto`, `Poisson`, `Rayleigh`, `ShiftedLoglogistic`, `StudentTNoncentral`, `StudentT`, `Triangular`, `Uniform`, `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`, `Pareto`, `Poisson`, `Rayleigh`, `ShiftedLoglogistic`, `StudentTNoncentral`, `StudentT`, `Triangular`, `Uniform`, `Weibull`, `WeightedDiscrete`