gkwreg (version 1.0.7)

rbeta_: Random Generation for the Beta Distribution (gamma, delta+1 Parameterization)

Description

Generates random deviates from the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by gamma (\(\gamma\)) and delta (\(\delta\)), corresponding to the standard Beta distribution with shape parameters shape1 = gamma and shape2 = delta + 1. This is a special case of the Generalized Kumaraswamy (GKw) distribution where \(\alpha = 1\), \(\beta = 1\), and \(\lambda = 1\).

Usage

rbeta_(n, gamma, delta)

Value

A numeric vector of length n containing random deviates from the Beta(\(\gamma, \delta+1\)) distribution, with values in (0, 1). The length of the result is determined by n and the recycling rule applied to the parameters (gamma, delta). Returns NaN if parameters are invalid (e.g., gamma <= 0, delta < 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.

gamma

First shape parameter (shape1), \(\gamma > 0\). Can be a scalar or a vector. Default: 1.0.

delta

Second shape parameter is delta + 1 (shape2), requires \(\delta \ge 0\) so that shape2 >= 1. Can be a scalar or a vector. Default: 0.0 (leading to shape2 = 1, i.e., Uniform).

Author

Lopes, J. E.

Details

This function generates samples from a Beta distribution with parameters shape1 = gamma and shape2 = delta + 1. It is equivalent to calling stats::rbeta(n, shape1 = gamma, shape2 = delta + 1).

This distribution arises as a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (rgkw) obtained by setting \(\alpha = 1\), \(\beta = 1\), and \(\lambda = 1\). It is therefore also equivalent to the McDonald (Mc)/Beta Power distribution (rmc) with \(\lambda = 1\).

The function likely calls R's underlying rbeta function but ensures consistent parameter recycling and handling within the C++ environment, matching the style of other functions in the related families.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd ed.). Wiley.

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation,

Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag.

See Also

rbeta (standard R implementation), rgkw (parent distribution random generation), rmc (McDonald/Beta Power random generation), dbeta_, pbeta_, qbeta_ (other functions for this parameterization, if they exist).

Examples

# \donttest{
set.seed(2030) # for reproducibility

# Generate 1000 samples using rbeta_
gamma_par <- 2.0 # Corresponds to shape1
delta_par <- 3.0 # Corresponds to shape2 - 1
shape1 <- gamma_par
shape2 <- delta_par + 1

x_sample <- rbeta_(1000, gamma = gamma_par, delta = delta_par)
summary(x_sample)

# Compare with stats::rbeta
x_sample_stats <- stats::rbeta(1000, shape1 = shape1, shape2 = shape2)
# Visually compare histograms or QQ-plots
hist(x_sample, main="rbeta_ Sample", freq=FALSE, breaks=30)
curve(dbeta_(x, gamma_par, delta_par), add=TRUE, col="red", lwd=2)
hist(x_sample_stats, main="stats::rbeta Sample", freq=FALSE, breaks=30)
curve(stats::dbeta(x, shape1, shape2), add=TRUE, col="blue", lwd=2)
# Compare summary stats (should be similar due to randomness)
print(summary(x_sample))
print(summary(x_sample_stats))

# Compare summary stats with rgkw(alpha=1, beta=1, lambda=1)
x_sample_gkw <- rgkw(1000, alpha = 1.0, beta = 1.0, gamma = gamma_par,
                     delta = delta_par, lambda = 1.0)
print("Summary stats for rgkw(a=1,b=1,l=1) sample:")
print(summary(x_sample_gkw))

# Compare summary stats with rmc(lambda=1)
x_sample_mc <- rmc(1000, gamma = gamma_par, delta = delta_par, lambda = 1.0)
print("Summary stats for rmc(l=1) sample:")
print(summary(x_sample_mc))

# }