Wald | R Documentation |

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

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).

Returns an R6 object inheriting from class SDistribution.

The distribution is supported on the Positive Reals.

Wald(mean = 1, shape = 1)

`quantile`

is
omitted as no closed form analytic expression could be found, decorate with `FunctionImputation`

for a numerical imputation.

Also known as the Inverse Normal distribution.

`distr6::Distribution`

-> `distr6::SDistribution`

-> `Wald`

`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.

`new()`

Creates a new instance of this R6 class.

Wald$new(mean = NULL, shape = NULL, decorators = NULL)

`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.

`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.

Wald$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).

Wald$mode(which = "all")

`which`

`(character(1) | numeric(1)`

Ignored if distribution is unimodal. Otherwise`"all"`

returns all modes, otherwise specifies which mode to return.

`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.

Wald$variance(...)

`...`

Unused.

`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.

Wald$skewness(...)

`...`

Unused.

`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.

Wald$kurtosis(excess = TRUE, ...)

`excess`

`(logical(1))`

If`TRUE`

(default) excess kurtosis returned.`...`

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.

Wald$mgf(t, ...)

`t`

`(integer(1))`

t integer to evaluate function at.`...`

Unused.

`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.

Wald$cf(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.

Wald$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.

Wald$clone(deep = FALSE)

`deep`

Whether to make a deep clone.

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`

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