Weibull {stats}

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 \ge 0. The cumulative distribution function is F(x) = 1 - \exp(-{(x/\sigma)}^a) on x \ge 0, the mean is E(X) = \sigma \Gamma(1 + 1/a), and the Var(X) = \sigma^2(\Gamma(1 + 2/a)-(\Gamma(1 + 1/a))^2).

Value

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.

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.

Source

[dpq]weibull are calculated directly from the definitions. rweibull uses inversion.

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 21. Wiley, New York.

See Also

The Exponential is a special case of the 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.tail=FALSE, log.p=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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