Weibull {Rlab}

The Weibull Distribution

Description

Density, distribution function, quantile function and random generation for the Weibull distribution with parameters alpha (or shape) and beta (or scale).

This special Rlab implementation allows the parameters alpha and beta to be used, to match the function description often found in textbooks.

Usage

dweibull(x, shape, scale = 1, alpha = shape, beta = scale, log = FALSE)
pweibull(q, shape, scale = 1, alpha = shape, beta = scale,
         lower.tail = TRUE, log.p = FALSE)
qweibull(p, shape, scale = 1, alpha = shape, beta = scale,
         lower.tail = TRUE, log.p = FALSE)
rweibull(n, shape, scale = 1, alpha = shape, beta = scale)

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.

alpha, beta

alpha and beta 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 alpha (or shape) parameter a and beta (or 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))^2).

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, alpha = 1), dexp(x))
all.equal(pweibull(x, alpha = 1, beta = 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, alpha = 1, beta = pi), qexp(x/11, rate = 1/pi))