KumGG: Kumaraswamy Generalized Gamma 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 KumGG distribution with parameters lambda, phi, tau, alpha and k.

dkumgg(x, tau, alpha = 1, k, lambda, phi, log = FALSE)

pkumgg(q, tau, alpha = 1, k, lambda, phi, lower.tail = TRUE,
  log.p = FALSE)

qkumgg(p, tau, alpha = 1, k, lambda, phi, lower.tail = TRUE,
  log.p = FALSE)

hkumgg(q, tau, alpha = 1, k, lambda, phi)

Hkumgg(q, tau, alpha = 1, k, lambda, phi)

rkumgg(n, tau, alpha = 1, k, lambda, phi, cens.prop = 0)

ml.kumgg(x, tau.ini, alpha.ini, k.ini, lambda.ini, phi.ini)

`x, q`	numeric vector of quantiles.
`alpha`	scale parameter α > 0.
`lambda`	shape parameter λ > 0.
`phi`	shape parameter φ ≥ 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.
`gamma`	shape parameter γ > 0.

The KumGG distribution was described by Ortega et al (2011) and has density

f(x) = (λφτ)/(αΓ[k])(x/α)^(τk-1)e^(-(x/α)^τ) (γ[k, (x/α)^τ])^(λ-1)(1-(γ[k, (x/α)^τ])^λ)^(φ-1)

where γ[., .] is the incomplete gamma ratio and Γ[.] is the gamma funcion. The scale parameter is α, the shape parameters are λ, φ and τ and k. The parameters λ and phi, come from the Kumaraswamy Generalized family introduced by Cordeiro and Castro (2011).

With phi = 1 KumGG becomes the Exponentiated Generalized Gamma distribution described by Cordeiro et al (2011). Additionally, if tau = k = 1 the Exponentiated Exponential.

When k = 1 then the KumGG distribution becomes the KumW distribution described by Cordeiro et al (2010). For k = 1 the KumGG becomes KumG distribution.

The above are arguably the most important sub-models of KumGG. More sub-models are described in Ortega et al (2011).

dkumgg gives the density, pkumgg gives the distribution function, qkumgg gives the quantile function, and rkumgg generates random values.

The length of the result is determined by n for rkumgg, 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.

ORTEGA, E. M. M.; CORDEIRO, G. M.; PASCOA, M. A. R. The generalized gamma geometric distribution. Journal of Statistical Theory and Applications, 2011, 10.3: 433-454.

CORDEIRO, G. M.; ORTEGA, E. M. M; SILVA, G. O. The exponentiated generalized gamma distribution with application to lifetime data. Journal of statistical computation and simulation, 2011, 81.7: 827-842.

CORDEIRO, G. M.; ORTEGA, E. M. M; NADARAJAH, S.. The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 2010, 347.8: 1399-1429.

CORDEIRO, G. M.; DE CASTRO, M. A new family of generalized distributions. Journal of statistical computation and simulation, 2011, 81.7: 883-898.

LINK TO OTHER PACKAGE DISTRIBUTIONS

# Generating values and comparing with the function
x <- rkumgg(10000, tau = 0.9, alpha = 2, k = 0.5, lambda = 3, phi = 10)
hist(x, probability = T, breaks = 100)
curve(dkumgg(x, tau = 0.9, alpha = 2, k = 0.5, lambda = 3, phi = 10),
      from = 0, to = 3, add = T)