reliaR (version 0.01)

ExpoWeibull: 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, $P[X > x]$.

Value

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

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.

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)