qekw {gkwreg} | R Documentation |
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)
Arguments
p |
Vector of probabilities (values between 0 and 1). |
alpha |
Shape parameter |
beta |
Shape parameter |
lambda |
Shape parameter |
lower_tail |
Logical; if |
log_p |
Logical; if |
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
.
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
forp = 0
(orp = -Inf
iflog_p = TRUE
, whenlower_tail = TRUE
). -
1
forp = 1
(orp = 0
iflog_p = TRUE
, whenlower_tail = TRUE
). -
NaN
forp < 0
orp > 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
.
Author(s)
Lopes, J. E.
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
# 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