Reverse Weibull Distribution

The Reverse Weibull distribution, also known as the Type III extreme value distribution, is the distribution of the negative of a Weibull-distributed random variable. It has an upper bound and a left-skewed density, and is parameterized by location, scale, and shape.

Usage

drevweibull(x, loc = 0, scale = 1, shape = 1, log = FALSE)

prevweibull(q, loc = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

qrevweibull(p, loc = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

rrevweibull(n, loc = 0, scale = 1, shape = 1)

dnweibull(x, loc = 0, scale = 1, shape = 1, log = FALSE)

pnweibull(q, loc = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

qnweibull(p, loc = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

rnweibull(n, loc = 0, scale = 1, shape = 1)

Arguments

x, q: vector of quantiles.
loc, scale, shape: location, scale and shape parameters (can be given as vectors).
log, log.p: logical; if TRUE, probabilities p are given as log(p) and the density is returned on the log scale.
lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p: vector of probabilities.
n: number of observations.

Value

drevweibull() and dnweibull() give the density function, prevweibull() and pnweibull() give the distribution function, qrevweibull() and qnweibull() give the quantile function, rrevweibull() and rnweibull() generate random deviates.

Details

Density function, distribution function, quantile function and random generation for the reverse (sometimes called negative) Weibull distribution with location, scale and shape parameters.

The reverse Weibull distribution function with parameters $`loc` = a$, $`scale` = b$ and $`shape` = s$ is $$G(z) = \exp\left\{-\left[-\left(\frac{z-a}{b}\right)\right]^s\right\}$$ for $z < a$ and one otherwise, where $b > 0$ and $s > 0$.

Note: Within extreme value theory the reverse Weibull distibution (also known as the negative Weibull distribution) is often referred to as the Weibull distribution. We make a distinction to avoid confusion with the three-parameter distribution used in survival analysis, which is related by a change of sign to the distribution given above.

Note

Based on code by Alec Stephenson previously published in the evd package, adapted to conform to package standards.

Examples


drevweibull(-5:-3, -1, 0.5, 0.8)
#> [1] 0.005386194 0.016885315 0.058502349
prevweibull(-5:-3, -1, 0.5, 0.8)
#> [1] 0.005102464 0.015101477 0.048246445
qrevweibull(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8)
#> [1] 1.969986 1.923317 1.862180 1.784071
rrevweibull(6, -1, 0.5, 0.8)
#> [1] -1.404639 -1.058543 -1.486602 -1.973033 -1.173827 -1.013722
p <- (1:9)/10
prevweibull(qrevweibull(p, -1, 2, 0.8), -1, 2, 0.8)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9