Quantile Function of the Kumaraswamy-Kumaraswamy (KKw) Distribution

Computes the quantile function (inverse CDF) for the Kumaraswamy-Kumaraswamy (KKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), delta ($\delta$), and lambda ($\lambda$). 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 $\gamma = 1$.

Usage

qkkw(
  p,
  alpha = 1,
  beta = 1,
  delta = 0,
  lambda = 1,
  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.
delta: Shape parameter delta >= 0. Can be a scalar or a vector. Default: 0.0.
lambda: Shape parameter lambda > 0. Can be a scalar or a vector. Default: 1.0.
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, delta, lambda). 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 KKw ($\gamma=1$) distribution is (see pkkw): $$ F(q) = 1 - \bigl\{1 - \bigl[1 - (1 - q^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta + 1} $$ Inverting this equation for $q$ yields the quantile function: $$ Q(p) = \left[ 1 - \left\{ 1 - \left[ 1 - (1 - p)^{1/(\delta+1)} \right]^{1/\lambda} \right\}^{1/\beta} \right]^{1/\alpha} $$ The function uses this closed-form expression and correctly handles the lower.tail and log.p arguments by transforming p appropriately before applying the formula.

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)
qkkw(p, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2)
#> [1] 0.1916721 0.4172992 0.6574119
# upper-tail quantiles
qkkw(p, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2, lower.tail = FALSE)
#> [1] 0.6574119 0.4172992 0.1916721

## qkkw() inverts pkkw()
all.equal(pkkw(qkkw(p, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2),
    alpha = 2, beta = 3, delta = 0.5, lambda = 1.2), p)
#> [1] TRUE