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 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 \le x], otherwise, P[X > x].

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))

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