dekw: Density of the Exponentiated Kumaraswamy (EKw) Distribution

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Density of the Exponentiated Kumaraswamy (EKw) Distribution

Description

Computes the probability density function (PDF) for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha (\alpha), beta (\beta), and lambda (\lambda). This distribution is defined on the interval (0, 1).

Usage

dekw(x, alpha, beta, lambda, log_prob = FALSE)

Arguments

x	Vector of quantiles (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.
log_prob	Logical; if TRUE, the logarithm of the density is returned (\log(f(x))). Default: FALSE.

Details

The probability density function (PDF) of the Exponentiated Kumaraswamy (EKw) distribution is given by:

f(x; \alpha, \beta, \lambda) = \lambda \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} \bigl[1 - (1 - x^\alpha)^\beta \bigr]^{\lambda - 1}

for 0 < x < 1.

The EKw distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (dgkw) obtained by setting the parameters \gamma = 1 and \delta = 0. When \lambda = 1, the EKw distribution reduces to the standard Kumaraswamy distribution.

Value

A vector of density values (f(x)) or log-density values (\log(f(x))). The length of the result is determined by the recycling rule applied to the arguments (x, alpha, beta, lambda). Returns 0 (or -Inf if log_prob = TRUE) for x outside the interval (0, 1), or NaN if parameters are invalid (e.g., alpha <= 0, beta <= 0, lambda <= 0).

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

dgkw (parent distribution density), pekw, qekw, rekw (other EKw functions),

Examples

# Example values
x_vals <- c(0.2, 0.5, 0.8)
alpha_par <- 2.0
beta_par <- 3.0
lambda_par <- 1.5 # Exponent parameter

# Calculate density
densities <- dekw(x_vals, alpha_par, beta_par, lambda_par)
print(densities)

# Calculate log-density
log_densities <- dekw(x_vals, alpha_par, beta_par, lambda_par, log_prob = TRUE)
print(log_densities)
# Check: should match log(densities)
print(log(densities))

# Compare with dgkw setting gamma = 1, delta = 0
densities_gkw <- dgkw(x_vals, alpha_par, beta_par, gamma = 1.0, delta = 0.0,
                      lambda = lambda_par)
print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero

# Plot the density for different lambda values
curve_x <- seq(0.01, 0.99, length.out = 200)
curve_y1 <- dekw(curve_x, alpha = 2, beta = 3, lambda = 0.5) # less peaked
curve_y2 <- dekw(curve_x, alpha = 2, beta = 3, lambda = 1.0) # standard Kw
curve_y3 <- dekw(curve_x, alpha = 2, beta = 3, lambda = 2.0) # more peaked

plot(curve_x, curve_y2, type = "l", main = "EKw Density Examples (alpha=2, beta=3)",
     xlab = "x", ylab = "f(x)", col = "red", ylim = range(0, curve_y1, curve_y2, curve_y3))
lines(curve_x, curve_y1, col = "blue")
lines(curve_x, curve_y3, col = "green")
legend("topright", legend = c("lambda=0.5", "lambda=1.0 (Kw)", "lambda=2.0"),
       col = c("blue", "red", "green"), lty = 1, bty = "n")

gkwreg documentation built on April 16, 2025, 1:10 a.m.