Cumulative Distribution Function (CDF) of the Beta Distribution (gamma, delta+1 Parameterization)

Computes the cumulative distribution function (CDF), \(F(q) = P(X \le q)\), for 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.

Usage

pbeta_(q, gamma = 1, delta = 0, lower.tail = TRUE, log.p = FALSE)

Arguments

q

Vector of quantiles (values generally between 0 and 1).

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).

lower.tail

Logical; if TRUE (default), probabilities are \(P(X \le q)\), otherwise, \(P(X > q)\).

log.p

Logical; if TRUE, probabilities \(p\) are given as \(\log(p)\). Default: FALSE.

Value

A vector of probabilities, \(F(q)\), or their logarithms/complements depending on lower.tail and log.p. The length of the result is determined by the recycling rule applied to the arguments (q, gamma, delta). When lower.tail = TRUE, returns 0 (or -Inf if log.p = TRUE) for q <= 0 and 1 (or 0 if log.p = TRUE) for q >= 1. 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. Boundary return values are adjusted accordingly for lower.tail = FALSE.

Details

This function computes the CDF of a Beta distribution with parameters shape1 = gamma and shape2 = delta + 1. It is equivalent to calling stats::pbeta(q, shape1 = gamma, shape2 = delta + 1, lower.tail = lower.tail, log.p = log.p).

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

The function likely calls R's underlying pbeta 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

See also

pbeta (standard R implementation), pgkw (parent distribution CDF), pmc (McDonald/Beta Power CDF), dbeta_, qbeta_, rbeta_.

Other cumulative distribution functions: pbkw(), pekw(), pgkw(), pkkw(), pkw(), pmc()

Author

Lopes, J. E.

Examples

q <- c(0.2, 0.5, 0.8)
pbeta_(q, gamma = 2, delta = 3)
#> [1] 0.26272 0.81250 0.99328
pbeta_(q, gamma = 2, delta = 3, lower.tail = FALSE)  # P(X > q)
#> [1] 0.73728 0.18750 0.00672
pbeta_(q, gamma = 2, delta = 3, log.p = TRUE)
#> [1] -1.336666453 -0.207639365 -0.006742681

## pbeta_() is the integral of dbeta_()
Fq <- pbeta_(0.5, gamma = 2, delta = 3)
all.equal(Fq, integrate(dbeta_, 0, 0.5, gamma = 2, delta = 3, rel.tol = 1e-10)$value)
#> [1] TRUE