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Create a Frechet distribution

Source: R/Frechet.R
Frechet.Rd

The Frechet distribution is a special case of the \link{GEV} distribution, obtained when the GEV shape parameter \(\xi\) is positive. It may be referred to as a type II extreme value distribution.

Usage

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

Arguments

location

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

Value

A Frechet 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 Frechet random variable with location parameter location = \(m\), scale parameter scale = \(s\), and shape parameter shape = \(\alpha\). A Frechet(\(m, s, \alpha\)) distribution is equivalent to a \link{GEV}(\(m + s, s / \alpha, 1 / \alpha\)) distribution.

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

Mean: \(m + s\Gamma(1 - 1/\alpha)\), for \(\alpha > 1\); undefined otherwise.

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

Variance: \(s^2 [\Gamma(1 - 2 / \alpha) - \Gamma(1 - 1 / \alpha)^2]\) for \(\alpha > 2\); undefined otherwise.

Probability density function (p.d.f):

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

Cumulative distribution function (c.d.f):

$$F(x) = \exp\{-[(x - m) / s] ^ {-\alpha} \}$$ for \(x > m\). The c.d.f. is 0 for \(x \leq m\).

See also

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

Examples


set.seed(27)

X <- Frechet(0, 2)
X
#> [1] "Frechet(location = 0, scale = 2, shape = 1)"

random(X, 10)
#>  [1] 69.7922625  0.8065071 14.8341823  1.8001889  1.3299308  2.1925530
#>  [7]  0.7621402  0.3326917  1.0064977  1.2115825

pdf(X, 0.7)
#> [1] 0.2344189
log_pdf(X, 0.7)
#> [1] -1.450646

cdf(X, 0.7)
#> [1] 0.05743262
quantile(X, 0.7)
#> [1] 5.607347

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