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
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.
Methods
Public methods
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Inherited methods
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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(...)
Arguments
...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(...)
Arguments
...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$skewness(...)
Arguments
...Unused.
Method kurtosis()
The kurtosis of a distribution is defined by the fourth standardised moment,
k_X = E_X[((x - μ)/σ)^4]
where E_X is the expectation of distribution X, μ is the mean of the
distribution and σ is the standard deviation of the distribution.
Excess Kurtosis is Kurtosis - 3.
Usage
Wald$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
deepWhether 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.
See Also
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
distr6 documentation built on March 28, 2022, 1:05 a.m.
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| 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
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.
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(...)
Arguments
...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(...)
Arguments
...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$skewness(...)
Arguments
...Unused.
Method kurtosis()
The kurtosis of a distribution is defined by the fourth standardised moment,
k_X = E_X[((x - μ)/σ)^4]
where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.
Usage
Wald$kurtosis(excess = TRUE, ...)
Arguments
excess(logical(1))
IfTRUE(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
deepWhether 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.
See Also
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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