gkwreg (version 1.0.7)

rekw: Random Number Generation for the Exponentiated Kumaraswamy (EKw) Distribution

Description

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, beta, lambda)

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

lambda

Shape parameter lambda > 0 (exponent parameter). Can be a scalar or a vector. Default: 1.0.

Author

Lopes, J. E.

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,

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).

See Also

rgkw (parent distribution random generation), dekw, pekw, qekw (other EKw functions), runif

Examples

# \donttest{
set.seed(2027) # for reproducibility

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

x_sample_ekw <- rekw(1000, alpha = alpha_par, beta = beta_par, lambda = lambda_par)
summary(x_sample_ekw)

# Histogram of generated values compared to theoretical density
hist(x_sample_ekw, breaks = 30, freq = FALSE, # freq=FALSE for density
     main = "Histogram of EKw Sample", xlab = "x", ylim = c(0, 3.0))
curve(dekw(x, alpha = alpha_par, beta = beta_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 <- qekw(prob_points, alpha = alpha_par, beta = beta_par,
                       lambda = lambda_par)
emp_quantiles <- quantile(x_sample_ekw, prob_points, type = 7)

plot(theo_quantiles, emp_quantiles, pch = 16, cex = 0.8,
     main = "Q-Q Plot for EKw 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, delta=0, ...)
# Note: individual values will differ due to randomness
x_sample_gkw <- rgkw(1000, alpha = alpha_par, beta = beta_par, gamma = 1.0,
                     delta = 0.0, lambda = lambda_par)
print("Summary stats for rekw sample:")
print(summary(x_sample_ekw))
print("Summary stats for rgkw(gamma=1, delta=0) sample:")
print(summary(x_sample_gkw)) # Should be similar

# }