# KumPar: Kumarasuammy Pareto Distribution In aneisse/survdistr: Survival Analysis Distributions

## Description

Density, distribution function, quantile function and random generation for the KumPar distribution with parameters `lambda`, `phi`, `beta` and `k`.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```dkumpar(x, beta, k, lambda, phi, log = FALSE) pkumpar(q, beta, k, lambda, phi, lower.tail = TRUE, log.p = FALSE) qkumpar(p, beta, k, lambda, phi, lower.tail = TRUE, log.p = FALSE) hkumpar(q, beta, k, lambda, phi) rkumpar(n, beta, k, lambda, phi, cens.prop = 0) ```

## Arguments

 `x, q` numeric vector of quantiles x > β. `beta` scale parameter β > 0. `k` shape parameter \k > 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 KumLL distribution was described by Pereira et al (2012) and has density

f(x) = (λφkβ^k)/(x^(k+1))(1-(β/x)^k)^(λ-1) (1-(1-(β/x)^k)^λ)^(φ-1)

for x > β and with scale parameter β, shape parameters λ, φ and k.

The parameters λ and phi, come from the Kumaraswamy Generalized family introduced by Cordeiro and Castro (2011).

With `phi = 1` KumPar becomes the Exponentiated Pareto distribution. In addition, when `lambda = 1` it becomes the Pareto distribution.

## Value

`dkumpar` gives the density, `pkumpar` gives the distribution function, `qkumpar` gives the quantile function, and `rkumpar` generates random values.

The length of the result is determined by `n` for `rkumpar`, 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

PEREIRA, M. B.; SILVA, R. B.; ZEA, L. M.; CORDEIRO, G. M. The kumaraswamy Pareto distribution. arXiv preprint arXiv:1204.1389, 2012.

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``` ```# Generating values and comparing with the function x <- rkumpar(10000, beta = 2, k = 0.5, lambda = 3, phi = 10) hist(x, probability = T, breaks = 100) curve(dkumpar(x, beta = 2, k = 0.5, lambda = 3, phi = 10), from = 2, to = 80, add = T) ```