GMWeibull: Generalized Modified Weibull Distribution
In aneisse/survdistr: Survival Analysis Distributions

Description Usage Arguments Details Value Author(s) Source References See Also Examples

Density, distribution function, quantile function and random generation for the GMWeibull distribution with parameters alpha, gamma, lambda and beta.

dgmweibull(x, alpha, gamma, lambda, beta, log = FALSE)

pgmweibull(q, alpha, gamma, lambda, beta, lower.tail = TRUE,
  log.p = FALSE)

qgmweibull(p, alpha, gamma, lambda, beta, lower.tail = TRUE,
  log.p = FALSE)

rgmweibull(n, alpha, gamma, lambda, beta, cens.prop = 0)

`x, q`	numeric vector of quantiles.
`alpha`	scale parameter α > 0.
`gamma`	shape parameter γ ≥ 0.
`lambda`	fragility factor parameter λ ≥ 0.
`beta`	shape parameter β > 0.
`log, log.p`	logical; if `TRUE`, probabilities/densities `p` are given as `log(p)`.
`lower.tail`	logical; if `TRUE`, probabilities are P[X ≤ x], otherwise, P[X ≥ x]
`n`	desired size of the random number sample.
`cens.prop`	proportion of censored data to be simulated. If greater than `0`, a matrix will be returned instead of a vector. The matrix will contain the random values and a censorship indicator variable.

The GMWeibull distribution was described by Carrasco et al (2008) and has density

f(x) = (α β x^(γ-1)(γ + λ x)e^(λ x - α x^γ e^(λ x)))/(1-e^(-α x^γ e^(λ x)))^(1-β)

with scale parameter α, shape parameters γ and β. The parameter λ, according to the authors, is a kind of accelerating factor working as a parameter of fragility in the survival of the individual as time increases.

With lambda = 0 and beta = 1 GMWeibull equals the classical two-parameter Weibull distribution. In addition, if gamma = 1 it equals the Exponential distribution and gamma = 2 it equals the Rayleigh distribution.

With gamma = 0 and beta = 1 GMWeibull equals the Extreme-value (log-gamma) distribution.

When lambda = 0 GMWeibull reduces to the Exponentiated Weibull distribution described by Mudholkar et al (1995). Furthermore, also setting gamma = 1 will reduce GMWeibbul to the exponentiated exponential distribution described by Gupta and Kundu (2001).

For beta = 1 GMWeibull will reduce to the Modified Weibull dsitribution introduced by Lai et al (2003).

dgmweibull gives the density, pgmweibull gives the distribution function, qgmweibull gives the quantile function, and rgmweibull generates random values.

The length of the result is determined by n for rgmweibull, for the other fucntions the length is the same as the vector passed to the first argument.

Only the first element of the logical arguments are used.

Anderson Neisse <a.neisse@gmail.com>

The source code of all distributions in this package can also be found on the survdistr Github repository.

CARRASCO, J. M. F.; ORTEGA, E. M. M.; CORDEIRO, G. M. A generalized modified Weibull distribution for lifetime modeling. Computational Statistics & Data Analysis, 2008, 53.2: 450-462.

MUDHOLKAR, G. S.; SRIVASTAVA, D. K.; FREIMER, M.. The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics, 1995, 37.4: 436-445.

LAI, C. D.; XIE, M.; MURTHY, D. N. P. A modified Weibull distribution. IEEE Transactions on reliability, 2003, 52.1: 33-37.

LINK TO OTHER PACKAGE DISTRIBUTIONS

# Equivalency with the Exponential distribution
all.equal(dgmweibull(5, alpha = 0.5, gamma = 1, lambda = 0, beta = 1),
          dexp(5, rate = 0.5))

# Generating values and comparing with the function
x <- rgmweibull(10000, alpha = 0.5, gamma = 3, lambda = 2, beta = 0.2)
hist(x, probability = T, breaks = 100)
curve(dgmweibull(x, alpha = 0.5, gamma = 3, lambda = 2, beta = 0.2),
      from = 0, to = 25, add = T)