The Reverse Weibull Distribution

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

dRevWeibull(x, loc=0, scale=1, shape=1, log = FALSE)
pRevWeibull(q, loc=0, scale=1, shape=1, lower.tail = TRUE)
qRevWeibull(p, loc=0, scale=1, shape=1, lower.tail = TRUE)
rRevWeibull(n, loc=0, scale=1, shape=1)

dNegWeibull(x, loc=0, scale=1, shape=1, log = FALSE)
pNegWeibull(q, loc=0, scale=1, shape=1, lower.tail = TRUE)
qNegWeibull(p, loc=0, scale=1, shape=1, lower.tail = TRUE)
rNegWeibull(n, loc=0, scale=1, shape=1)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

loc, scale, shape

Location, scale and shape parameters (can be given as vectors).

log

Logical; if TRUE, the log density is returned.

lower.tail

Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

Details

The reverse (or negative) 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.

Value

dRevWeibull and dNegWeibull give the density function,

pRevWeibull and pNegWeibull give the distribution function,

qRevWeibull and qNegWeibull give the quantile function,

rRevWeibull and rNegWeibull generate random deviates.

See also

rFrechet, rGenExtrVal, rGumbel

Author

Alec Stephenson <alec_stephenson@hotmail.com>

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] -1.092336 -3.983379 -1.197627 -1.653867 -1.066734 -1.302922
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