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 > 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 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.

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.

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 4.2.0 Index]