RevWeibull {distributions3}R Documentation

Create a reversed Weibull distribution

Description

The reversed (or negated) Weibull distribution is a special case of the ⁠\link{GEV}⁠ distribution, obtained when the GEV shape parameter ξ\xi is negative. It may be referred to as a type III extreme value distribution.

Usage

RevWeibull(location = 0, scale = 1, shape = 1)

Arguments

location

The location (maximum) parameter mm. location can be any real number. Defaults to 0.

scale

The scale parameter ss. scale can be any positive number. Defaults to 1.

shape

The scale parameter α\alpha. shape can be any positive number. Defaults to 1.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let XX be a reversed Weibull random variable with location parameter location = mm, scale parameter scale = ss, and shape parameter shape = α\alpha. An RevWeibull(m,s,αm, s, \alpha) distribution is equivalent to a ⁠\link{GEV}⁠(ms,s/α,1/αm - s, s / \alpha, -1 / \alpha) distribution.

If XX has an RevWeibull(m,λ,km, \lambda, k) distribution then mXm - X has a ⁠\link{Weibull}⁠(k,λk, \lambda) distribution, that is, a Weibull distribution with shape parameter kk and scale parameter λ\lambda.

Support: (,m)(-\infty, m).

Mean: m+sΓ(1+1/α)m + s\Gamma(1 + 1/\alpha).

Median: m+s(ln2)1/αm + s(\ln 2)^{1/\alpha}.

Variance: s2[Γ(1+2/α)Γ(1+1/α)2]s^2 [\Gamma(1 + 2 / \alpha) - \Gamma(1 + 1 / \alpha)^2].

Probability density function (p.d.f):

f(x)=αs1[(xm)/s]α1exp{[(xm)/s]α}f(x) = \alpha s ^ {-1} [-(x - m) / s] ^ {\alpha - 1}% \exp\{-[-(x - m) / s] ^ {\alpha} \}

for x<mx < m. The p.d.f. is 0 for xmx \geq m.

Cumulative distribution function (c.d.f):

F(x)=exp{[(xm)/s]α}F(x) = \exp\{-[-(x - m) / s] ^ {\alpha} \}

for x<mx < m. The c.d.f. is 1 for xmx \geq m.

Value

A RevWeibull object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples


set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

[Package distributions3 version 0.2.1 Index]