reliaR (version 0.01)

MOEW: The Marshall-Olkin Extended Weibull (MOEW) distribution

Description

Density, distribution function, quantile function and random generation for the Marshall-Olkin Extended Weibull (MOEW) distribution with tilt parameter alpha and scale parameter lambda.

Usage

dmoew(x, alpha, lambda, log = FALSE)
pmoew(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qmoew(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rmoew(n, alpha, lambda)

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.

lambda

tilt 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

dmoew gives the density, pmoew gives the distribution function, qmoew gives the quantile function, and rmoew generates random deviates.

Details

The Marshall-Olkin extended Weibull (MOEW) distribution has density $$f(x) = \frac{{\lambda \alpha x^{\alpha - 1} \exp\left({-x^\alpha}\right)}}{{\left\{{1 - (1 - \lambda)\;\exp\left({-x^\alpha}\right)}\right\}^2}};\, x > 0, \lambda > 0, \alpha > 0$$ where $\alpha$ and $\lambda$ are the tilt and scale parameters, respectively.

References

Marshall, A. W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the Weibull and Weibull families. Biometrika,84(3):641-652.

Marshall, A. W., Olkin, I.(2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer, New York.

See Also

.Random.seed about random number; smoew for MOEW survival / hazard etc. functions;

Examples

## Load data sets
data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.3035937,  lambda.est = 279.2177754

dmoew(sys2, 0.3035937, 279.2177754, log = FALSE)
pmoew(sys2, 0.3035937, 279.2177754, lower.tail = TRUE, log.p = FALSE)
qmoew(0.25, 0.3035937, 279.2177754, lower.tail=TRUE, log.p = FALSE)
rmoew(50, 0.3035937, 279.2177754)