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

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 [e-mail]

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

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 May 9, 2019, 10:37 a.m.