rgkw: Generalized Kumaraswamy Distribution Random Generation

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Generalized Kumaraswamy Distribution Random Generation

Description

Generates random deviates from the five-parameter Generalized Kumaraswamy (GKw) distribution defined on the interval (0, 1).

Usage

rgkw(n, alpha, beta, gamma, 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.
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.
lambda	Shape parameter lambda > 0. Can be a scalar or a vector. Default: 1.0.

Details

The generation method relies on the transformation property: if V \sim \mathrm{Beta}(\gamma, \delta+1), then the random variable X defined as

X = \left\{ 1 - \left[ 1 - V^{1/\lambda} \right]^{1/\beta} \right\}^{1/\alpha}

follows the GKw(\alpha, \beta, \gamma, \delta, \lambda) distribution.

The algorithm proceeds as follows:

1. Generate V from stats::rbeta(n, shape1 = gamma, shape2 = delta + 1).

2. Calculate v = V^{1/\lambda}.

3. Calculate w = (1 - v)^{1/\beta}.

4. Calculate x = (1 - w)^{1/\alpha}.

Parameters (alpha, beta, gamma, delta, lambda) are recycled to match the length required by n. Numerical stability is maintained by handling potential edge cases during the transformations.

Value

A vector of length n containing random deviates from the GKw distribution. The length of the result is determined by n and the recycling rule applied to the parameters (alpha, beta, gamma, delta, lambda). Returns NaN if parameters are invalid (e.g., alpha <= 0, beta <= 0, gamma <= 0, delta < 0, lambda <= 0).

Author(s)

Lopes, J. E.

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.

See Also

dgkw, pgkw, qgkw, rbeta, set.seed

Examples

set.seed(1234) # for reproducibility

# Generate 1000 random values from a specific GKw distribution (Kw case)
x_sample <- rgkw(1000, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1)
summary(x_sample)

# Histogram of generated values compared to theoretical density
hist(x_sample, breaks = 30, freq = FALSE, # freq=FALSE for density scale
     main = "Histogram of GKw(2,3,1,0,1) Sample", xlab = "x", ylim = c(0, 2.5))
curve(dgkw(x, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1),
      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 <- qgkw(prob_points, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1)
emp_quantiles <- quantile(x_sample, prob_points)

plot(theo_quantiles, emp_quantiles, pch = 16, cex = 0.8,
     main = "Q-Q Plot for GKw(2,3,1,0,1)",
     xlab = "Theoretical Quantiles", ylab = "Empirical Quantiles (n=1000)")
abline(a = 0, b = 1, col = "blue", lty = 2)

# Using vectorized parameters: generate 1 value for each alpha
alphas_vec <- c(0.5, 1.0, 2.0)
n_param <- length(alphas_vec)
samples_vec <- rgkw(n_param, alpha = alphas_vec, beta = 2, gamma = 1, delta = 0, lambda = 1)
print(samples_vec) # One sample for each alpha value
# Result length matches n=3, parameters alpha recycled accordingly

# Example with invalid parameters (should produce NaN)
invalid_sample <- rgkw(1, alpha = -1, beta = 2, gamma = 1, delta = 0, lambda = 1)
print(invalid_sample)

gkwreg documentation built on April 16, 2025, 1:10 a.m.