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

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, delta, 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).
NaN for invalid parameters (e.g., gamma <= 0, delta < 0).

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,

Author

Lopes, J. E.

Examples

# \donttest{
# Example values
p_vals <- c(0.1, 0.5, 0.9)
gamma_par <- 2.0 # Corresponds to shape1
delta_par <- 3.0 # Corresponds to shape2 - 1
shape1 <- gamma_par
shape2 <- delta_par + 1

# Calculate quantiles using qbeta_
quantiles <- qbeta_(p_vals, gamma_par, delta_par)
print(quantiles)
#> [1] 0.1122350 0.3138102 0.5838904

# Compare with stats::qbeta
quantiles_stats <- stats::qbeta(p_vals, shape1 = shape1, shape2 = shape2)
print(paste("Max difference vs stats::qbeta:", max(abs(quantiles - quantiles_stats))))
#> [1] "Max difference vs stats::qbeta: 0"

# Compare with qgkw setting alpha=1, beta=1, lambda=1
quantiles_gkw <- qgkw(p_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par,
                      delta = delta_par, lambda = 1.0)
print(paste("Max difference vs qgkw:", max(abs(quantiles - quantiles_gkw))))
#> [1] "Max difference vs qgkw: 5.55111512312578e-17"

# Compare with qmc setting lambda=1
quantiles_mc <- qmc(p_vals, gamma = gamma_par, delta = delta_par, lambda = 1.0)
print(paste("Max difference vs qmc:", max(abs(quantiles - quantiles_mc))))
#> [1] "Max difference vs qmc: 0"

# Calculate quantiles for upper tail
quantiles_upper <- qbeta_(p_vals, gamma_par, delta_par, lower_tail = FALSE)
print(quantiles_upper)
#> [1] 0.5838904 0.3138102 0.1122350
print(stats::qbeta(p_vals, shape1, shape2, lower.tail = FALSE))
#> [1] 0.5838904 0.3138102 0.1122350

# Calculate quantiles from log probabilities
log_p_vals <- log(p_vals)
quantiles_logp <- qbeta_(log_p_vals, gamma_par, delta_par, log_p = TRUE)
print(quantiles_logp)
#> [1] 0.1122350 0.3138102 0.5838904
print(stats::qbeta(log_p_vals, shape1, shape2, log.p = TRUE))
#> [1] 0.1122350 0.3138102 0.5838904

# Verify inverse relationship with pbeta_
p_check <- 0.75
q_calc <- qbeta_(p_check, gamma_par, delta_par)
p_recalc <- pbeta_(q_calc, gamma_par, delta_par)
print(paste("Original p:", p_check, " Recalculated p:", p_recalc))
#> [1] "Original p: 0.75  Recalculated p: 0.75"
# abs(p_check - p_recalc) < 1e-9 # Should be TRUE

# Boundary conditions
print(qbeta_(c(0, 1), gamma_par, delta_par)) # Should be 0, 1
#> [1] 0 1
print(qbeta_(c(-Inf, 0), gamma_par, delta_par, log_p = TRUE)) # Should be 0, 1
#> [1] 0 1

# }