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Create a reversed Weibull distribution

Source: R/ReversedWeibull.R
RevWeibull.Rd

The reversed (or negated) Weibull distribution is a special case of the \link{GEV} distribution, obtained when the GEV shape parameter \(\xi\) 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 \(m\). location can be any real number. Defaults to 0.

scale

The scale parameter \(s\). scale can be any positive number. Defaults to 1.

shape

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

Value

A RevWeibull object.

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 \(X\) be a reversed Weibull random variable with location parameter location = \(m\), scale parameter scale = \(s\), and shape parameter shape = \(\alpha\). An RevWeibull(\(m, s, \alpha\)) distribution is equivalent to a \link{GEV}(\(m - s, s / \alpha, -1 / \alpha\)) distribution.

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

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

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

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

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

Probability density function (p.d.f):

$$f(x) = \alpha s ^ {-1} [-(x - m) / s] ^ {\alpha - 1}% \exp\{-[-(x - m) / s] ^ {\alpha} \}$$ for \(x < m\). The p.d.f. is 0 for \(x \geq m\).

Cumulative distribution function (c.d.f):

$$F(x) = \exp\{-[-(x - m) / s] ^ {\alpha} \}$$ for \(x < m\). The c.d.f. is 1 for \(x \geq m\).

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
#> [1] "RevWeibull(location = 1, scale = 2, shape = 1)"

random(X, 10)
#>  [1]   0.9426871  -3.9596589   0.7303525  -1.2219891  -2.0076752  -0.8243573
#>  [7]  -4.2483783 -11.0231439  -2.9741769  -2.3014673

pdf(X, 0.7)
#> [1] 0.430354
log_pdf(X, 0.7)
#> [1] -0.8431472

cdf(X, 0.7)
#> [1] 0.860708
quantile(X, 0.7)
#> [1] 0.2866501

cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 0.7))
#> [1] 0.7