Quantile Function of the Exponentiated Kumaraswamy (EKw) Distribution

Computes the quantile function (inverse CDF) for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), 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 $\gamma = 1$ and $\delta = 0$.

Usage

qekw(p, alpha, beta, lambda, 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.
lambda: Shape parameter lambda > 0 (exponent parameter). 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, 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).
NaN for invalid parameters (e.g., alpha <= 0, beta <= 0, lambda <= 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 EKw ($\gamma=1, \delta=0$) distribution is $F(q) = [1 - (1 - q^\alpha)^\beta ]^\lambda$ (see pekw). Inverting this equation for $q$ yields the quantile function: $$ Q(p) = \left\{ 1 - \left[ 1 - p^{1/\lambda} \right]^{1/\beta} \right\}^{1/\alpha} $$ 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. This is equivalent to the general GKw quantile function (qgkw) evaluated with $\gamma=1, \delta=0$.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

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 <- 3.0
lambda_par <- 1.5

# Calculate quantiles
quantiles <- qekw(p_vals, alpha_par, beta_par, lambda_par)
print(quantiles)
#> [1] 0.2787375 0.5311017 0.7695287

# Calculate quantiles for upper tail probabilities P(X > q) = p
quantiles_upper <- qekw(p_vals, alpha_par, beta_par, lambda_par,
                        lower_tail = FALSE)
print(quantiles_upper)
#> [1] 0.7695287 0.5311017 0.2787375
# Check: qekw(p, ..., lt=F) == qekw(1-p, ..., lt=T)
print(qekw(1 - p_vals, alpha_par, beta_par, lambda_par))
#> [1] 0.7695287 0.5311017 0.2787375

# Calculate quantiles from log probabilities
log_p_vals <- log(p_vals)
quantiles_logp <- qekw(log_p_vals, alpha_par, beta_par, lambda_par,
                       log_p = TRUE)
print(quantiles_logp)
#> [1] 0.2787375 0.5311017 0.7695287
# Check: should match original quantiles
print(quantiles)
#> [1] 0.2787375 0.5311017 0.7695287

# Compare with qgkw setting gamma = 1, delta = 0
quantiles_gkw <- qgkw(p_vals, alpha = alpha_par, beta = beta_par,
                     gamma = 1.0, delta = 0.0, lambda = lambda_par)
print(paste("Max difference:", max(abs(quantiles - quantiles_gkw)))) # Should be near zero
#> [1] "Max difference: 1.11022302462516e-16"

# Verify inverse relationship with pekw
p_check <- 0.75
q_calc <- qekw(p_check, alpha_par, beta_par, lambda_par)
p_recalc <- pekw(q_calc, alpha_par, beta_par, lambda_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(qekw(c(0, 1), alpha_par, beta_par, lambda_par)) # Should be 0, 1
#> [1] 0 1
print(qekw(c(-Inf, 0), alpha_par, beta_par, lambda_par, log_p = TRUE)) # Should be 0, 1
#> [1] 0 1
# }