KumW: Kumarasuammy 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 KumW distribution with parameters lambda, phi, c, k and s.

dkumw(x, beta, c, lambda, phi, log = FALSE)

pkumw(q, beta, c, lambda, phi, lower.tail = TRUE, log.p = FALSE)

qkumw(p, beta, c, lambda, phi, lower.tail = TRUE, log.p = FALSE)

hkumw(q, beta, c, lambda, phi)

rkumw(n, beta, c, lambda, phi, cens.prop = 0)

mlkumw(x, b.ini, c.ini, l.ini, p.ini)

`x, q`	numeric vector of quantiles.
`beta`	scale parameter β > 0.
`c`	shape parameter c > 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.

The KumW distribution was described by Cordeiro et al (2010) and has density

f(x) = λφcβ^cx^(x-1)e^(-(βx)^c)(1-e^(-(βx)^c))^(λ-1) (1-(1-e^(-(βx)^c))^λ)^(φ-1)

with scale parameter β, shape parameters λ, φ and c. The parameters λ and phi, come from the Kumaraswamy Generalized family introduced by Cordeiro and Castro (2011).

The KumW is a special case of KumBII introduced by Parna<c3><ad>ba et al (2013).

With phi = 1 KumW becomes the Exponentiated Weibull distribution. In addition, when lambda = 1 it becomes the Weibull distribution.

When phi = c = 1 then the KumW distribution becomes the exponentiated exponential distribution .

The above are arguably the most important sub-models to KumW. More su-models are decribed by Cordeiro et al (2010) as well as some expasions for the KumW pdf.

dkumw gives the density, pkumw gives the distribution function, qkumw gives the quantile function, and rkumw generates random values.

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

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.

PARANA<c3><8d>BA, P. F.; Ortega, E. M.; Cordeiro, G. M.; Pascoa, M. A. D. The Kumaraswamy Burr XII distribution: theory and practice. Journal of Statistical Computation and Simulation, 2013, 83.11: 2117-2143.

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

# Equivalency with the Weibull
all.equal(dkumw(5, beta = 0.5, c = 2, lambda = 1, phi = 1),
          dweibull(5, shape = 2, scale = 1/0.5))

# Generating values and comparing with the function
x <- rkumw(10000, beta = 1.5, c = 0.5, lambda = 3, phi = 10)
hist(x, probability = T, breaks = 100)
curve(dkumw(x, beta = 1.5, c = 0.5, lambda = 3, phi = 10),
      from = 0, to = 2, add = T)