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

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, beta, gamma, delta)

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). Returns NaN if parameters are invalid (e.g., alpha <= 0, beta <= 0, gamma <= 0, delta < 0).

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:

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

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

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

Author

Lopes, J. E.

Examples

# \donttest{
set.seed(2026) # for reproducibility

# Generate 1000 random values from a specific BKw distribution
alpha_par <- 2.0
beta_par <- 1.5
gamma_par <- 1.0
delta_par <- 0.5

x_sample_bkw <- rbkw(1000, alpha = alpha_par, beta = beta_par,
                     gamma = gamma_par, delta = delta_par)
summary(x_sample_bkw)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.01187 0.36143 0.51194 0.51019 0.67906 0.99278 

# Histogram of generated values compared to theoretical density
hist(x_sample_bkw, breaks = 30, freq = FALSE, # freq=FALSE for density
     main = "Histogram of BKw Sample", xlab = "x", ylim = c(0, 2.5))
curve(dbkw(x, alpha = alpha_par, beta = beta_par, gamma = gamma_par,
           delta = delta_par),
      add = TRUE, col = "red", lwd = 2, n = 201)
legend("topright", legend = "Theoretical PDF", col = "red", lwd = 2, bty = "n")


# Comparing empirical and theoretical quantiles (Q-Q plot)
prob_points <- seq(0.01, 0.99, by = 0.01)
theo_quantiles <- qbkw(prob_points, alpha = alpha_par, beta = beta_par,
                       gamma = gamma_par, delta = delta_par)
emp_quantiles <- quantile(x_sample_bkw, prob_points, type = 7)

plot(theo_quantiles, emp_quantiles, pch = 16, cex = 0.8,
     main = "Q-Q Plot for BKw Distribution",
     xlab = "Theoretical Quantiles", ylab = "Empirical Quantiles (n=1000)")
abline(a = 0, b = 1, col = "blue", lty = 2)


# Compare summary stats with rgkw(..., lambda=1, ...)
# Note: individual values will differ due to randomness
x_sample_gkw <- rgkw(1000, alpha = alpha_par, beta = beta_par, gamma = gamma_par,
                     delta = delta_par, lambda = 1.0)
print("Summary stats for rbkw sample:")
#> [1] "Summary stats for rbkw sample:"
print(summary(x_sample_bkw))
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.01187 0.36143 0.51194 0.51019 0.67906 0.99278 
print("Summary stats for rgkw(lambda=1) sample:")
#> [1] "Summary stats for rgkw(lambda=1) sample:"
print(summary(x_sample_gkw)) # Should be similar
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.04279 0.36298 0.52400 0.51632 0.67744 0.98445 

# }