KumLL: Kumarasuammy Log-Logistic 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 KumLL distribution with parameters lambda, phi, c, k and s.

dkumll(x, alpha, gamma, lambda, phi, log = FALSE)

pkumll(q, alpha, gamma, lambda, phi, lower.tail = TRUE, log.p = FALSE)

qkumll(p, alpha, gamma, lambda, phi, lower.tail = TRUE, log.p = FALSE)

hkumll(q, alpha, gamma, lambda, phi)

rkumll(n, alpha, gamma, lambda, phi, cens.prop = 0)

mlkumll(x, a.ini, g.ini, l.ini, p.ini)

`x, q`	numeric vector of quantiles.
`alpha`	scale parameter α > 0.
`gamma`	shape parameter γ > 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 KumLL distribution was described by Santana et al (2012) and has density

f(x) = (λφγ)/(α^(λγ))x^(λγ-1) (1+(t/α)^γ)^(-λ-1)(1-(1-1/(1+(t/α)^γ))^λ)^(φ-1)

with scale parameter α, shape parameters λ, φ and γ that govern the distribution's skewness. The parameters λ and phi, come from the Kumaraswamy Generalized family introduced by Cordeiro and Castro (2011).

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

With phi = 1 KumLL becomes the Exponentiated Log-Logistic distribution. In addition, when lambda = 1 it becomes the Log-Logistic distribution. Those are arguably the most important sub-models to KumLL.

When lambda = 1 then the KumLL distribution becomes the BXII distribution described by Zimmer et al (1998).

This distribution's failure rate function accommodates increasing, decreasing, unimodal and bathtub shaped forms, that depend basically on the values of the shape parameters. Moreover, it is quite flexible for modeling survival data.

dkumll gives the density, pkumll gives the distribution function, qkumll gives the quantile function, and rkumll generates random values.

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

DE SANTANA, T. V. F.; Ortega, E. M.; Cordeiro, G. M.; Silva, G. O. The Kumaraswamy-log-logistic distribution. Journal of Statistical Theory and Applications, 2012, 11.3: 265-291.

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.

ZIMMER, W. J.; KEATS, J. B.; WANG, F. K. The Burr XII distribution in reliability analysis. Journal of quality technology, 1998, 30.4: 386-394.

LINK TO OTHER PACKAGE DISTRIBUTIONS

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