# moeev: Marshall-Olkin extended extreme value (or Weibull)... In Rumenick/flexcure: Flexible Parametric Survival Models with Cure Fraction

## Description

Density, distribution function, quantile function and random generation for the Marshall-Olkin extended extreme value distribution (MOEEV) distribution with location parameter mu, scale parameter sigma and tilt paramater alpha.

## Usage

 ```1 2 3 4``` ```dmoeev(x, mu = 0, sigma = 1, alpha, log = FALSE) pmoeev(q, mu = 0, sigma = 1, alpha, lower.tail = TRUE, log.p = FALSE) qmoeev(p, mu = 0, sigma = 1, alpha, lower.tail = TRUE, log.p = FALSE) rmoeev(n, mu = 0, sigma = 1, alpha) ```

## Arguments

 `x, q` vector of quantiles. `mu` vector of location parameters. `sigma` vector of scale parameters. `alpha` vector of tilt parameters. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `p` vector of probabilities. `n` number of observations. If length(n) > 1, the length is taken to be the number required. `lower.tail` logical; if TRUE (default), probabilities are P(X ≤ x), otherwise, P(X > x).

## Details

The Marshall-Olkin Extended Extreme Value (MOEEV) distribution has density

f(x;μ,σ,α) = {α exp[w-exp(w)]} / {σ x{1-(1-α)exp[-exp(w)]}^2}, x>0

where w=[log(x)-μ]/σ and -∞<μ<∞, σ>0 and α>0 are the location, scale and tilt parameters, respectively. If α = 1, we obtain the extreme value distribution.

Consider the parameterisations used by `dexp` and `dweibull`. If μ = -log(rate) and σ = 1, we obtain the Marshall-Olkin Extended Exponential (MOEE) distribution. In the case, that μ = log(scale) and σ = 1/shape, we obtain the Marshall-Olkin Extended Weibull (MOEW) distribution. With the above definitions,

if MOEE: `dmoeev(x, mu = -log(rate), sigma = 1, alpha) = dmoee(x, mu, alpha)`

if MOEW: `dmoeev(x, mu = log(scale), sigma = 1/shape, alpha)`

The Marshall-Olkin extended extreme value distribution simplifies to the exponential and Weibull distributions with the following parameterisations:

 `dmoeev(x, mu, sigma = 1, alpha = 1)` ` = ` `dexp(x, rate = 1/exp(mu))` `dmoeev(x, mu, sigma, alpha = 1)` ` = ` `dweibull(x, shape=1/sigma, scale=exp(mu))`

## Value

`dmoeev` gives the density, `pmoeev` gives the distribution function, `qmoeev` gives the quantile function, and `rmoeev` generates random deviates.

## Author(s)

Rumenick Pereira da Silva [email protected]

## 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.

## Examples

 ```1 2 3 4``` ```x <- rmoeev(1000, mu = 2, sigma = 1, alpha = 1) all.equal(dmoeev(x, mu = 2, sigma = 1, alpha = 1), dexp(x, rate = 1/exp(2))) x <- rmoeev(1000, mu = 2, sigma = 2, alpha = 1) all.equal(dmoeev(x, mu = 2, sigma = 2, alpha = 1), dweibull(x, shape=1/2, scale = exp(2))) ```

Rumenick/flexcure documentation built on July 9, 2018, 2:20 p.m.