| Weibull {stats} | R Documentation |
Density, distribution function, quantile function and random
generation for the Weibull distribution with parameters shape
and scale.
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)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If length(n) > 1, the length
is taken to be the number required. |
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 P[X <= x], otherwise, P[X > x]. |
The Weibull distribution with shape parameter a and
scale parameter b has density given by
f(x) = (a/b) (x/b)^(a-1) exp(- (x/b)^a)
for x >= 0. The cumulative distribution function is F(x) = 1 - exp(- (x/b)^a) on x >= 0, the mean is E(X) = b Gamma(1 + 1/a), and the Var(X) = b^2 * (Gamma(1 + 2/a) - (Gamma(1 + 1/a))^2).
dweibull gives the density,
pweibull gives the distribution function,
qweibull gives the quantile function, and
rweibull generates random deviates.
Invalid arguments will result in return value NaN, with a warning.
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.
[dpq]weibull are calculated directly from the definitions.
rweibull uses inversion.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 21. Wiley, New York.
The Exponential is a special case of the Weibull distribution.
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))