qekw: Quantile Function of the Exponentiated Kumaraswamy (EKw) Distribution

Description

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 = 1, beta = 1, lambda = 1, lower.tail = TRUE, log.p = 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.

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.

Author

Lopes, J. E.

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.

Examples

Run this code

# \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)

# 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)
# Check: qekw(p, ..., lt=F) == qekw(1-p, ..., lt=T)
print(qekw(1 - p_vals, alpha_par, beta_par, lambda_par))

# 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)
# Check: should match original quantiles
print(quantiles)

# 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

# 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))
# 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
print(qekw(c(-Inf, 0), alpha_par, beta_par, lambda_par, log.p = TRUE)) # Should be 0, 1
# }

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