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

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 = 1, 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).

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). An out-of-bound or missing parameter is an error, not a return value: the wrapper stops with a message naming the parameter. An infinite parameter is not currently intercepted there and reaches the C++ layer, which treats it as invalid.

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, 81(7), 883-898. doi:10.1080/00949650903530745

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 random generation functions: rbkw(), rekw(), rgkw(), rkkw(), rkw(), rmc()

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rbeta_(1000, gamma = 2, delta = 3)
summary(x)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.01114 0.20168 0.31755 0.33709 0.45444 0.90972 

## The sample follows the distribution
hist(x, breaks = 30, freq = FALSE, main = "")
curve(dbeta_(x, gamma = 2, delta = 3), add = TRUE)

ks.test(x, pbeta_, gamma = 2, delta = 3)
#> 
#> 	Asymptotic one-sample Kolmogorov-Smirnov test
#> 
#> data:  x
#> D = 0.026144, p-value = 0.5013
#> alternative hypothesis: two-sided
#>