| % File src/library/stats/man/Weibull.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{Weibull} |
| \alias{Weibull} |
| \alias{dweibull} |
| \alias{pweibull} |
| \alias{qweibull} |
| \alias{rweibull} |
| \title{The Weibull Distribution} |
| \description{ |
| Density, distribution function, quantile function and random |
| generation for the Weibull distribution with parameters \code{shape} |
| and \code{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{ |
| \item{x, q}{vector of quantiles.} |
| \item{p}{vector of probabilities.} |
| \item{n}{number of observations. If \code{length(n) > 1}, the length |
| is taken to be the number required.} |
| \item{shape, scale}{shape and scale parameters, the latter defaulting to 1.} |
| \item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).} |
| \item{lower.tail}{logical; if TRUE (default), probabilities are |
| \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.} |
| } |
| \value{ |
| \code{dweibull} gives the density, |
| \code{pweibull} gives the distribution function, |
| \code{qweibull} gives the quantile function, and |
| \code{rweibull} generates random deviates. |
| |
| Invalid arguments will result in return value \code{NaN}, with a warning. |
| |
| The length of the result is determined by \code{n} for |
| \code{rweibull}, and is the maximum of the lengths of the |
| numerical arguments for the other functions. |
| |
| The numerical arguments other than \code{n} are recycled to the |
| length of the result. Only the first elements of the logical |
| arguments are used. |
| } |
| \details{ |
| The Weibull distribution with \code{shape} parameter \eqn{a} and |
| \code{scale} parameter \eqn{\sigma}{b} has density given by |
| \deqn{f(x) = (a/\sigma) {(x/\sigma)}^{a-1} \exp (-{(x/\sigma)}^{a})}{f(x) = (a/b) (x/b)^(a-1) exp(- (x/b)^a)} for \eqn{x > 0}. |
| The cumulative distribution function is |
| \eqn{F(x) = 1 - \exp(-{(x/\sigma)}^a)}{F(x) = 1 - exp(- (x/b)^a)} |
| on \eqn{x > 0}, the |
| mean is \eqn{E(X) = \sigma \Gamma(1 + 1/a)}{E(X) = b \Gamma(1 + 1/a)}, and |
| the \eqn{Var(X) = \sigma^2(\Gamma(1 + 2/a)-(\Gamma(1 + 1/a))^2)}{Var(X) = b^2 * (\Gamma(1 + 2/a) - (\Gamma(1 + 1/a))^2)}. |
| } |
| \note{ |
| The cumulative hazard \eqn{H(t) = - \log(1 - F(t))}{H(t) = - log(1 - F(t))} |
| is |
| \preformatted{-pweibull(t, a, b, lower = FALSE, log = TRUE) |
| } |
| which is just \eqn{H(t) = {(t/b)}^a}{H(t) = (t/b)^a}. |
| } |
| \source{ |
| \code{[dpq]weibull} are calculated directly from the definitions. |
| \code{rweibull} uses inversion. |
| } |
| \references{ |
| Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) |
| \emph{Continuous Univariate Distributions}, volume 1, chapter 21. |
| Wiley, New York. |
| } |
| \seealso{ |
| \link{Distributions} for other standard distributions, including |
| the \link{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)) |
| } |
| \keyword{distribution} |