Random Number Generation for the Kumaraswamy (Kw) Distribution

Generates random deviates from the two-parameter Kumaraswamy (Kw) distribution with shape parameters alpha ($\alpha$) and beta ($\beta$).

Usage

rkw(n, alpha = 1, beta = 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.

Value

A vector of length n containing random deviates from the Kw distribution, with values in (0, 1). The length of the result is determined by n and the recycling rule applied to the parameters (alpha, beta). 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 Kw distribution, where $Q(p)$ is the Kw quantile function (qkw): $$ Q(p) = \left\{ 1 - (1 - p)^{1/\beta} \right\}^{1/\alpha} $$ The implementation generates $U$ using runif and applies this transformation. This is equivalent to the general GKw generation method (rgkw) evaluated at $\gamma=1, \delta=0, \lambda=1$.

References

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

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81. doi:10.1016/j.stamet.2008.04.001

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 <- rkw(1000, alpha = 2, beta = 3)
summary(x)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.01246 0.30481 0.44835 0.45516 0.60611 0.95701 

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

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