gkwreg (version 1.0.7)

rgkw: 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)

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).

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.

Author

Lopes, J. E.

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.

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

# \donttest{
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)
# }