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Random Number Generation for the Beta-Kumaraswamy (BKw) Distribution

Source: R/bkw.R
rbkw.Rd

Generates random deviates from the Beta-Kumaraswamy (BKw) distribution with parameters alpha (\(\alpha\)), beta (\(\beta\)), gamma (\(\gamma\)), and delta (\(\delta\)). This distribution is a special case of the Generalized Kumaraswamy (GKw) distribution where the parameter \(\lambda = 1\).

Usage

rbkw(n, alpha = 1, beta = 1, gamma = 1, delta = 0)

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.

Value

A vector of length n containing random deviates from the BKw distribution. The length of the result is determined by n and the recycling rule applied to the parameters (alpha, beta, gamma, delta). 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 relationship between the GKw distribution and the Beta distribution. The general procedure for GKw (rgkw) is: If \(W \sim \mathrm{Beta}(\gamma, \delta+1)\), then \(X = \{1 - [1 - W^{1/\lambda}]^{1/\beta}\}^{1/\alpha}\) follows the GKw(\(\alpha, \beta, \gamma, \delta, \lambda\)) distribution.

For the BKw distribution, \(\lambda=1\). Therefore, the algorithm simplifies to:

  1. Generate \(V \sim \mathrm{Beta}(\gamma, \delta+1)\) using rbeta.

  2. Compute the BKw variate \(X = \{1 - (1 - V)^{1/\beta}\}^{1/\alpha}\).

This procedure is implemented efficiently, handling parameter recycling as needed.

References

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).

See also

rgkw (parent distribution random generation), dbkw, pbkw, qbkw (other BKw functions), rbeta

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

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rbkw(1000, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)
summary(x)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.04977 0.34173 0.45853 0.46130 0.57929 0.93144 

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

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