qbeta_ {gkwreg} | R Documentation |
Quantile Function of the Beta Distribution (gamma, delta+1 Parameterization)
Description
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 ( |
delta |
Second shape parameter is |
lower_tail |
Logical; if |
log_p |
Logical; if |
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.
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
forp = 0
(orp = -Inf
iflog_p = TRUE
, whenlower_tail = TRUE
). -
1
forp = 1
(orp = 0
iflog_p = TRUE
, whenlower_tail = TRUE
). -
NaN
forp < 0
orp > 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
.
Author(s)
Lopes, J. E.
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,
See Also
qbeta
(standard R implementation),
qgkw
(parent distribution quantile function),
qmc
(McDonald/Beta Power quantile function),
dbeta_
, pbeta_
, rbeta_
(other functions for this parameterization, if they exist).
Examples
# 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)
# 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))))
# 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))))
# 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))))
# Calculate quantiles for upper tail
quantiles_upper <- qbeta_(p_vals, gamma_par, delta_par, lower_tail = FALSE)
print(quantiles_upper)
print(stats::qbeta(p_vals, shape1, shape2, lower.tail = FALSE))
# 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)
print(stats::qbeta(log_p_vals, shape1, shape2, log.p = TRUE))
# 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))
# 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
print(qbeta_(c(-Inf, 0), gamma_par, delta_par, log_p = TRUE)) # Should be 0, 1