Quantile Function of the Beta-Kumaraswamy (BKw) Distribution

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.

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.

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.

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.

Author

Lopes, J. E.

Examples

# \donttest{
# 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)
#> [1] 0.2138865 0.5149104 0.8003866

# 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)
#> [1] 0.8003866 0.5149104 0.2138865
# Check: qbkw(p, ..., lt=F) == qbkw(1-p, ..., lt=T)
print(qbkw(1 - p_vals, alpha_par, beta_par, gamma_par, delta_par))
#> [1] 0.8003866 0.5149104 0.2138865

# 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)
#> [1] 0.2138865 0.5149104 0.8003866
# Check: should match original quantiles
print(quantiles)
#> [1] 0.2138865 0.5149104 0.8003866

# 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
#> [1] "Max difference: 2.77555756156289e-17"

# 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))
#> [1] "Original p: 0.75  Recalculated p: 0.75"
# 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
#> [1] 0 1
print(qbkw(c(-Inf, 0), alpha_par, beta_par, gamma_par, delta_par, log_p = TRUE)) # Should be 0, 1
#> [1] 0 1

# }