# KumW: Kumarasuammy Weibull Distribution In aneisse/survdistr: Survival Analysis Distributions

## Description

Density, distribution function, quantile function and random generation for the KumW distribution with parameters `lambda`, `phi`, `c`, `k` and `s`.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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) ```

## Arguments

 `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.

## Details

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.

## Value

`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.

## Author(s)

Anderson Neisse <[email protected]>

## Source

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

## References

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.

 ```1 2 3 4 5 6 7 8 9``` ```# 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) ```