Random Number Generation for the Generalized Kumaraswamy Distribution

Generates random deviates from the five-parameter Generalized Kumaraswamy (GKw) distribution defined on the interval (0, 1).

Usage

rgkw(n, alpha = 1, beta = 1, gamma = 1, delta = 0, lambda = 1)

Arguments

n

Number of observations. If length(n) > 1, the length is taken to be the number required. Must be a non-negative integer.

alpha

Shape parameter alpha > 0. Can be a scalar or a vector. Default: 1.0.

beta

Shape parameter beta > 0. Can be a scalar or a vector. Default: 1.0.

gamma

Shape parameter gamma > 0. Can be a scalar or a vector. Default: 1.0.

delta

Shape parameter delta >= 0. Can be a scalar or a vector. Default: 0.0.

lambda

Shape parameter lambda > 0. Can be a scalar or a vector. Default: 1.0.

Value

A vector of length n containing random deviates from the GKw distribution. The length of the result is determined by n and the recycling rule applied to the parameters (alpha, beta, gamma, delta, lambda). An out-of-bound or missing parameter is an error, not a return value: the wrapper stops with a message naming the parameter. An infinite parameter is not currently intercepted there and reaches the C++ layer, which treats it as invalid.

Details

The generation method relies on the transformation property: if \(V \sim \mathrm{Beta}(\gamma, \delta+1)\), then the random variable X defined as $$ X = \left\{ 1 - \left[ 1 - V^{1/\lambda} \right]^{1/\beta} \right\}^{1/\alpha} $$ follows the GKw(\(\alpha, \beta, \gamma, \delta, \lambda\)) distribution.

The algorithm proceeds as follows:

1. Generate V from stats::rbeta(n, shape1 = gamma, shape2 = delta + 1).

2. Calculate \(v = V^{1/\lambda}\).

3. Calculate \(w = (1 - v)^{1/\beta}\).

4. Calculate \(x = (1 - w)^{1/\alpha}\).

Parameters (alpha, beta, gamma, delta, lambda) are recycled to match the length required by n. Numerical stability is maintained by handling potential edge cases during the transformations.

References

Carrasco, J. M. F., Ferrari, S. L. P., & Cordeiro, G. M. (2010). A new generalized Kumaraswamy distribution. arXiv preprint arXiv:1004.0911. doi:10.48550/arXiv.1004.0911

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898. doi:10.1080/00949650903530745

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88. doi:10.1016/0022-1694(80)90036-0

See also

dgkw, pgkw, qgkw, rbeta, set.seed

Other random generation functions: rbeta_(), rbkw(), rekw(), rkkw(), rkw(), rmc()

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rgkw(1000, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)
summary(x)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.07502 0.38251 0.49412 0.49439 0.60747 0.93562 

## The sample follows the distribution
hist(x, breaks = 30, freq = FALSE, main = "")
curve(dgkw(x, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2),
    add = TRUE)

ks.test(x, pgkw, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)
#> 
#> 	Asymptotic one-sample Kolmogorov-Smirnov test
#> 
#> data:  x
#> D = 0.024673, p-value = 0.5766
#> alternative hypothesis: two-sided
#>