The Weibull (Shape-Decay) Distribution

Description

Density, distribution function, quantile function, random generation, expected value, variance, median, and raw moments for the shape-decay parameterization of the Weibull distribution.

Usage

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dweisd(x, shape, decay, log=FALSE)
pweisd(q, shape, decay, lower.tail=TRUE, log.p=FALSE)
qweisd(p, shape, decay, lower.tail=TRUE, log.p=FALSE)
rweisd(n, shape, decay)
eweisd(shape, decay)
vweisd(shape, decay)
medweisd(shape, decay)
rmomweisd(r, shape, decay)

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, decay

shape and decay parameters. See Details.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

r

the rth raw moment is returned. See Value.

Details

The weisd distribution with shape parameter s and decay parameter d has density

f(x) = s*d*x^(s-1) * exp(-d*x^s)

for x >= 0; s, d > 0. Compared to the usual shape-scale parameterization used in dweibull etc., with shape parameter a and scale parameter b, the relationship between the parameters is given by a = s and b = d^(-1/s), or equivalently, s = a and d = b^(-a).

Value

dweisd gives the density, pweisd gives the distribution function, qweisd gives the quantile function, and rweisd generates random deviates. eweisd gives the expected value, vweisd gives the variance, and medweisd gives the median for the specified parameters. rmomweisd gives the rth raw moment for the specified parameters; for example, rmomweisd(1, 2, 3) is equivalent to eweisd(2, 3).

Note

The distribution function, using the parameters given in Details, has the very simple form

F(x) = 1 - exp(-d*x^s)

from which other functions such as the hazard function are easily derived.

Author(s)

Daniel Dvorkin

See Also

mvweisd for the multivariate version; thetahat for parameter estimation; Weibull in package stats.

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