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/densities are given as logarithms.

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

Value

dweibull gives the density, pweibull is the cumulative distribution function, and qweibull is the quantile function of the Weibull distribution. rweibull generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is determined by n for rweibull, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

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

Distributions for other standard distributions, including the Exponential which 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, tolerance = 1e-15)
all.equal(qweibull(x/11, shape = 1, scale = pi), qexp(x/11, rate = 1/pi))

[Package stats version 4.6.0 Index]