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.
RevWeibull(location = 0, scale = 1, shape = 1)
A RevWeibull
object.
The location (maximum) parameter \(m\).
location
can be any real number. Defaults to 0
.
The scale parameter \(s\).
scale
can be any positive number. Defaults to 1
.
The scale parameter \(\alpha\).
shape
can be any positive number. Defaults to 1
.
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 \(x \geq m\).
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))
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