Random Number Generation for the Exponentiated Kumaraswamy (EKw) Distribution

Generates random deviates from the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), and lambda ($\lambda$). This distribution is a special case of the Generalized Kumaraswamy (GKw) distribution where $\gamma = 1$ and $\delta = 0$.

Usage

rekw(n, alpha = 1, beta = 1, 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.
lambda: Shape parameter lambda > 0 (exponent parameter). Can be a scalar or a vector. Default: 1.0.

Value

A vector of length n containing random deviates from the EKw distribution. The length of the result is determined by n and the recycling rule applied to the parameters (alpha, beta, 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 uses the inverse transform (quantile) method. That is, if $U$ is a random variable following a standard Uniform distribution on (0, 1), then $X = Q(U)$ follows the EKw distribution, where $Q(u)$ is the EKw quantile function (qekw): $$ Q(u) = \left\{ 1 - \left[ 1 - u^{1/\lambda} \right]^{1/\beta} \right\}^{1/\alpha} $$ This is computationally equivalent to the general GKw generation method (rgkw) when specialized for $\gamma=1, \delta=0$, as the required Beta(1, 1) random variate is equivalent to a standard Uniform(0, 1) variate. The implementation generates $U$ using runif and applies the transformation above.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

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

Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag. (General methods for random variate generation).

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rekw(1000, alpha = 2, beta = 3, lambda = 1.2)
summary(x)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.02361 0.34658 0.48448 0.48738 0.63249 0.95960 

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

ks.test(x, pekw, alpha = 2, beta = 3, lambda = 1.2)
#> 
#> 	Asymptotic one-sample Kolmogorov-Smirnov test
#> 
#> data:  x
#> D = 0.014051, p-value = 0.9891
#> alternative hypothesis: two-sided
#>