qbkw: Quantile Function of the Beta-Kumaraswamy (BKw) Distribution

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Quantile Function of the Beta-Kumaraswamy (BKw) Distribution

Description

Computes the quantile function (inverse CDF) for the Beta-Kumaraswamy (BKw) distribution with parameters alpha (\alpha), beta (\beta), gamma (\gamma), and delta (\delta). It finds the value q such that P(X \le q) = p. This distribution is a special case of the Generalized Kumaraswamy (GKw) distribution where the parameter \lambda = 1.

Usage

qbkw(p, alpha, beta, gamma, delta, lower_tail = TRUE, log_p = FALSE)

Arguments

p	Vector of probabilities (values between 0 and 1).
alpha	Shape parameter alpha > 0. Can be a scalar or a vector. Default: 1.0.
beta	Shape parameter beta > 0. Can be a scalar or a vector. Default: 1.0.
gamma	Shape parameter gamma > 0. Can be a scalar or a vector. Default: 1.0.
delta	Shape parameter delta >= 0. Can be a scalar or a vector. Default: 0.0.
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.

Details

The quantile function Q(p) is the inverse of the CDF F(q). The CDF for the BKw (\lambda=1) distribution is F(q) = I_{y(q)}(\gamma, \delta+1), where y(q) = 1 - (1 - q^\alpha)^\beta and I_z(a,b) is the regularized incomplete beta function (see pbkw).

To find the quantile q, we first invert the outer Beta part: let y = I^{-1}_{p}(\gamma, \delta+1), where I^{-1}_p(a,b) is the inverse of the regularized incomplete beta function, computed via qbeta. Then, we invert the inner Kumaraswamy part: y = 1 - (1 - q^\alpha)^\beta, which leads to q = \{1 - (1-y)^{1/\beta}\}^{1/\alpha}. Substituting y gives the quantile function:

Q(p) = \left\{ 1 - \left[ 1 - I^{-1}_{p}(\gamma, \delta+1) \right]^{1/\beta} \right\}^{1/\alpha}

The function uses this formula, calculating I^{-1}_{p}(\gamma, \delta+1) via qbeta(p, gamma, delta + 1, ...) while respecting the lower_tail and log_p arguments.

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, alpha, beta, 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., alpha <= 0, beta <= 0, gamma <= 0, delta < 0).

Boundary return values are adjusted accordingly for lower_tail = FALSE.

Author(s)

Lopes, J. E.

References

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

See Also

qgkw (parent distribution quantile function), dbkw, pbkw, rbkw (other BKw functions), qbeta

Examples

# Example values
p_vals <- c(0.1, 0.5, 0.9)
alpha_par <- 2.0
beta_par <- 1.5
gamma_par <- 1.0
delta_par <- 0.5

# Calculate quantiles
quantiles <- qbkw(p_vals, alpha_par, beta_par, gamma_par, delta_par)
print(quantiles)

# Calculate quantiles for upper tail probabilities P(X > q) = p
quantiles_upper <- qbkw(p_vals, alpha_par, beta_par, gamma_par, delta_par,
                        lower_tail = FALSE)
print(quantiles_upper)
# Check: qbkw(p, ..., lt=F) == qbkw(1-p, ..., lt=T)
print(qbkw(1 - p_vals, alpha_par, beta_par, gamma_par, delta_par))

# Calculate quantiles from log probabilities
log_p_vals <- log(p_vals)
quantiles_logp <- qbkw(log_p_vals, alpha_par, beta_par, gamma_par, delta_par,
                       log_p = TRUE)
print(quantiles_logp)
# Check: should match original quantiles
print(quantiles)

# Compare with qgkw setting lambda = 1
quantiles_gkw <- qgkw(p_vals, alpha_par, beta_par, gamma = gamma_par,
                     delta = delta_par, lambda = 1.0)
print(paste("Max difference:", max(abs(quantiles - quantiles_gkw)))) # Should be near zero

# Verify inverse relationship with pbkw
p_check <- 0.75
q_calc <- qbkw(p_check, alpha_par, beta_par, gamma_par, delta_par)
p_recalc <- pbkw(q_calc, alpha_par, beta_par, 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(qbkw(c(0, 1), alpha_par, beta_par, gamma_par, delta_par)) # Should be 0, 1
print(qbkw(c(-Inf, 0), alpha_par, beta_par, gamma_par, delta_par, log_p = TRUE)) # Should be 0, 1

gkwreg documentation built on April 16, 2025, 1:10 a.m.