Random Number Generation for the kkw Distribution

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

Usage

rkkw(n, alpha, beta, delta, lambda)

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.
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 kkw distribution. The length of the result is determined by n and the recycling rule applied to the parameters (alpha, beta, delta, lambda). Returns NaN if parameters are invalid (e.g., alpha <= 0, beta <= 0, delta < 0, lambda <= 0).

Details

The generation method uses the inverse transform method based on the quantile function (qkkw). The kkw quantile function is: $$ Q(p) = \left[ 1 - \left\{ 1 - \left[ 1 - (1 - p)^{1/(\delta+1)} \right]^{1/\lambda} \right\}^{1/\beta} \right]^{1/\alpha} $$ Random deviates are generated by evaluating $Q(p)$ where $p$ is a random variable following the standard Uniform distribution on (0, 1) (runif).

This is equivalent to the general method for the GKw distribution (rgkw) specialized for $\gamma=1$. The GKw method generates $W \sim \mathrm{Beta}(\gamma, \delta+1)$ and then applies transformations. When $\gamma=1$, $W \sim \mathrm{Beta}(1, \delta+1)$, which can be generated via $W = 1 - V^{1/(\delta+1)}$ where $V \sim \mathrm{Unif}(0,1)$. Substituting this $W$ into the GKw transformation yields the same result as evaluating $Q(1-V)$ above (noting $p = 1-V$ is also Uniform).

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(2025) # for reproducibility

# Generate 1000 random values from a specific kkw distribution
alpha_par <- 2.0
beta_par <- 3.0
delta_par <- 0.5
lambda_par <- 1.5

x_sample_kkw <- rkkw(1000, alpha = alpha_par, beta = beta_par,
                       delta = delta_par, lambda = lambda_par)
summary(x_sample_kkw)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.06135 0.34367 0.46565 0.46855 0.58892 0.88108 

# Histogram of generated values compared to theoretical density
hist(x_sample_kkw, breaks = 30, freq = FALSE, # freq=FALSE for density
     main = "Histogram of kkw Sample", xlab = "x", ylim = c(0, 3.5))
curve(dkkw(x, alpha = alpha_par, beta = beta_par, delta = delta_par,
            lambda = lambda_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 <- qkkw(prob_points, alpha = alpha_par, beta = beta_par,
                        delta = delta_par, lambda = lambda_par)
emp_quantiles <- quantile(x_sample_kkw, prob_points, type = 7) # type 7 is default

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


# Compare summary stats with rgkw(..., gamma=1, ...)
# Note: individual values will differ due to randomness
x_sample_gkw <- rgkw(1000, alpha = alpha_par, beta = beta_par, gamma = 1.0,
                     delta = delta_par, lambda = lambda_par)
print("Summary stats for rkkw sample:")
#> [1] "Summary stats for rkkw sample:"
print(summary(x_sample_kkw))
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.06135 0.34367 0.46565 0.46855 0.58892 0.88108 
print("Summary stats for rgkw(gamma=1) sample:")
#> [1] "Summary stats for rgkw(gamma=1) sample:"
print(summary(x_sample_gkw)) # Should be similar
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.04101 0.34241 0.46850 0.46428 0.58476 0.88441 
# }