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

Source: R/beta.R
qbeta_.Rd

Computes the quantile function (inverse CDF) for the standard Beta distribution, using a parameterization common in generalized distribution families. It finds the value q such that \(P(X \le q) = p\). 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

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

Arguments

p

Vector of probabilities (values 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 \(p = P(X \le q)\), otherwise, probabilities are \(p = P(X > q)\).

log.p

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

Value

A vector of quantiles corresponding to the given probabilities p. The length of the result is determined by the recycling rule applied to the arguments (p, gamma, delta). Returns:

  • 0 for p = 0 (or p = -Inf if log.p = TRUE, when lower.tail = TRUE).

  • 1 for p = 1 (or p = 0 if log.p = TRUE, when lower.tail = TRUE).

  • NaN for p < 0 or p > 1 (or corresponding log scale).

  • 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 quantiles of a Beta distribution with parameters shape1 = gamma and shape2 = delta + 1. It is equivalent to calling stats::qbeta(p, 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 (qgkw) obtained by setting \(\alpha = 1\), \(\beta = 1\), and \(\lambda = 1\). It is therefore also equivalent to the McDonald (Mc)/Beta Power distribution (qmc) with \(\lambda = 1\).

The function likely calls R's underlying qbeta function but ensures consistent parameter recycling and handling within the C++ environment, matching the style of other functions in the related families. Boundary conditions (p=0, p=1) are handled explicitly.

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

qbeta (standard R implementation), qgkw (parent distribution quantile function), qmc (McDonald/Beta Power quantile function), dbeta_, pbeta_, rbeta_.

Other quantile functions: qbkw(), qekw(), qgkw(), qkkw(), qkw(), qmc()

Author

Lopes, J. E.

Examples

p <- c(0.1, 0.5, 0.9)
qbeta_(p, gamma = 2, delta = 3)
#> [1] 0.1122350 0.3138102 0.5838904
qbeta_(p, gamma = 2, delta = 3, lower.tail = FALSE)  # upper-tail quantiles
#> [1] 0.5838904 0.3138102 0.1122350

## qbeta_() inverts pbeta_()
all.equal(pbeta_(qbeta_(p, gamma = 2, delta = 3), gamma = 2, delta = 3), p)
#> [1] TRUE