| Weibull {base} | R Documentation |
The Weibull Distribution
Description
Density, distribution function, quantile function and random
generation for the Weibull distribution with parameters shape
and scale.
Usage
dweibull(x, shape, scale = 1, log = FALSE)
pweibull(q, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
qweibull(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
rweibull(n, shape, scale = 1)
Arguments
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
shape, scale |
shape and scale parameters, the latter defaulting to 1. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
Details
The Weibull distribution with shape parameter a and
scale parameter \sigma has density given by
f(x) = (a/\sigma) {(x/\sigma)}^{a-1} \exp (-{(x/\sigma)}^{a})
for x > 0.
The cumulative is
F(x) = 1 - \exp(-{(x/\sigma)}^a), the
mean is E(X) = \sigma \Gamma(1 + 1/a), and
the Var(X) = \sigma^2(\Gamma(1 + 2/a)-\Gamma(1 + 1/a)).
Value
dweibull gives the density,
pweibull gives the distribution function,
qweibull gives the quantile function, and
rweibull generates random deviates.
Note
The cumulative hazard H(t) = - \log(1 - F(t))
is -pweibull(t, a, b, lower = FALSE, log = TRUE) which is just
H(t) = {(t/b)}^a.
See Also
dexp for the Exponential which is a special case of a
Weibull distribution.
Examples
x <- c(0,rlnorm(50))
all.equal(dweibull(x, shape = 1), dexp(x))
all.equal(pweibull(x, shape = 1, scale = pi), pexp(x, rate = 1/pi))
## Cumulative hazard H():
all.equal(pweibull(x, 2.5, pi, lower=FALSE, log=TRUE), -(x/pi)^2.5, tol=1e-15)
all.equal(qweibull(x/11, shape = 1, scale = pi), qexp(x/11, rate = 1/pi))