gkwreg (version 1.0.7)

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, beta, lambda, 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.

See Also

qgkw (parent distribution quantile function), dekw, pekw, rekw (other EKw functions), qunif

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)

# 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
# }