The Weibull Distribution.

Description

Density, distribution, quantile, random number generation, and parameter estimation functions for the Weibull distribution with parameters shape and scale. Parameter estimation can be based on a weighted or unweighted i.i.d sample and can be carried out analytically or numerically.

Usage

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dWeibull(x, shape = 2, scale = 2, params = list(shape = 2, scale = 2))

pWeibull(q, shape = 2, scale = 2, params = list(shape = 2, scale = 2))

qWeibull(p, shape = 2, scale = 2, params = list(shape = 2, scale = 2))

rWeibull(n, shape = 2, scale = 2, params = list(shape = 2, scale = 2))

eWeibull(X, w, method = c("numerical.MLE", "moments"), ...)

lWeibull(X, w, shape = 2, scale = 2, params = list(shape = 2, scale = 2),
  logL = TRUE)

Arguments

x,q

A vector of quantiles.

shape

Shape parameter.

scale

Scale parameter.

params

A list that includes all named parameters

p

A vector of probabilities.

n

Number of observations.

X

Sample observations.

w

An optional vector of sample weights.

method

Parameter estimation method.

...

Additional parameters.

logL

logical; if TRUE, lWeibull gives the log-likelihood, otherwise the likelihood is given.

Details

The Weibull distribution is a special case of the generalised gamma distribution. The dWeibull(), pWeibull(), qWeibull(),and rWeibull() functions serve as wrappers of the standard dgamma, pgamma, qgamma, and rgamma functions with in the stats package. They allow for the parameters to be declared not only as individual numerical values, but also as a list so parameter estimation can be carried out.

The Weibull distribution with parameters shape=a and scale=b has probability density function,

f(x)= (a/b)(x/b)^{a-1}exp(-(x/b)^a)

for x >0. Parameter estimation can be carried out using the method of moments as done by Winston (2003) or by numerical maximum likelihood estimation.

The log-likelihood function of the Weibull distribution is given by

l(a,b|x) = n(ln a - ln b) + (a-1)∑ ln(xi/b) - ∑(xi/b)^{a}

The score function and information matrix are as given by Rinne (p.412).

Value

dWeibull gives the density, pWeibull the distribution function, qWeibull the quantile function, rWeibull generates random deviates, and eWeibull estimates the distribution parameters. lWeibull provides the log-likelihood function.

Author(s)

Haizhen Wu and A. Jonathan R. Godfrey.
Updates and bug fixes by Sarah Pirikahu.

References

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

Rinne, H. (2009) The Weibull Distribution A Handbook, chapter 11, Chapman & Hall/CRC.

Winston, W.L (2003) Operations Research: Applications and algorithms, 4th Ed, Duxbury.

See Also

ExtDist for other standard distributions.

Examples

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# Parameter estimation for a distribution with known shape parameters
X <- rWeibull(n=1000, params=list(shape=1.5, scale=0.5))
est.par <- eWeibull(X=X, method="numerical.MLE"); est.par
plot(est.par)

#  Fitted density curve and histogram
den.x <- seq(min(X),max(X),length=100)
den.y <- dWeibull(den.x,shape=est.par$shape,scale=est.par$scale)
hist(X, breaks=10, col="red", probability=TRUE, ylim = c(0,1.1*max(den.y)))
lines(den.x, den.y, col="blue", lwd=2)   # Original data
lines(density(X), lty=2)                 # Fitted curve

# Extracting shape and scale parameters
est.par[attributes(est.par)$par.type=="shape"]
est.par[attributes(est.par)$par.type=="scale"]

# Parameter Estimation for a distribution with unknown shape parameters
# Example from: Rinne (2009) Dataset p.338 and example pp.418-419
# Parameter estimates are given as shape = 99.2079 and scale = 2.5957. The log-likelihood
# for this data and Rinne's parameter estimates is -1163.278.
data <- c(35,38,42,56,58,61,63,76,81,83,86,90,99,104,113,114,117,119,141,183)
est.par <- eWeibull(X=data, method="numerical.MLE"); est.par
plot(est.par)

# Estimates calculated by eWeibull differ from those given by Rinne(2009).
# However, eWeibull's parameter estimates appear to be an improvement, due to a larger
# log-likelihood of -99.09037 (as given by lWeibull below).

 # log-likelihood function
lWeibull(data, param = est.par)

# evaluate the precision of estimation by Hessian matrix
H <- attributes(est.par)$nll.hessian
var <- solve(H)
se <- sqrt(diag(var));se