# Density of the Exponentiated Kumaraswamy (EKw) Distribution 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). ```r 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`. ## Returns 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`). ## 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. ## Examples ```r # 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") ``` ## 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), ## Author(s) Lopes, J. E.

dekw function