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 = 1,
  beta = 1,
  gamma = 1,
  delta = 0,
  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).
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

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, 81(7), 883-898. doi:10.1080/00949650903530745

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88. doi:10.1016/0022-1694(80)90036-0

Author

Lopes, J. E.

Examples

p <- c(0.1, 0.5, 0.9)
qbkw(p, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)
#> [1] 0.2348264 0.4542020 0.6790672
# upper-tail quantiles
qbkw(p, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lower.tail = FALSE)
#> [1] 0.6790672 0.4542020 0.2348264

## qbkw() inverts pbkw()
all.equal(pbkw(qbkw(p, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5),
    alpha = 2, beta = 3, gamma = 1.5, delta = 0.5), p)
#> [1] TRUE