ExpoWeibull: The Exponentiated Weibull(EW) distribution

In reliaR: Comprehensive Tools for some Probability Distributions

The Exponentiated Weibull(EW) distribution

Description

Density, distribution function, quantile function and random generation for the Exponentiated Weibull(EW) distribution with shape parameters alpha and theta.

Usage

dexpo.weibull(x, alpha, theta, log = FALSE)
pexpo.weibull(q, alpha, theta, lower.tail = TRUE, log.p = FALSE)
qexpo.weibull(p, alpha, theta, lower.tail = TRUE, log.p = FALSE)
rexpo.weibull(n, alpha, theta)

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.
alpha	shape parameter.
theta	shape parameter.
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 Exponentiated Weibull(EW) distribution has density

f(x; \alpha, \theta) = \alpha \; \theta \; x^{\alpha - 1} \; e^{-x^{\alpha}} \left\{1-\exp \left(-x^{\alpha}\right)\right\}^{\theta -1};\; (\alpha, \theta) > 0, x > 0

where \alpha and \theta are the shape and scale parameters, respectively.

Value

dexpo.weibull gives the density, pexpo.weibull gives the distribution function, qexpo.weibull gives the quantile function, and rexpo.weibull generates random deviates.

References

Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42(2), 299-302.

Murthy, D.N.P., Xie, M. and Jiang, R. (2003). Weibull Models, Wiley, New York.

Nassar, M.M., and Eissa, F. H. (2003). On the Exponentiated Weibull Distribution, Communications in Statistics - Theory and Methods, 32(7), 1317-1336.

See Also

.Random.seed about random number; sexpo.weibull for Exponentiated Weibull(EW) survival / hazard etc. functions

Examples

## Load data sets
data(stress)
## Maximum Likelihood(ML) Estimates of alpha & theta for the data(stress)
## Estimates of alpha & theta using 'maxLik' package
## alpha.est =1.026465, theta.est = 7.824943

dexpo.weibull(stress, 1.026465, 7.824943, log = FALSE)
pexpo.weibull(stress, 1.026465, 7.824943, lower.tail = TRUE, log.p = FALSE)
qexpo.weibull(0.25, 1.026465, 7.824943, lower.tail=TRUE, log.p = FALSE)
rexpo.weibull(30, 1.026465, 7.824943)

reliaR documentation built on Nov. 5, 2025, 6:57 p.m.