# Density of the Kumaraswamy-Kumaraswamy (kkw) Distribution Computes the probability density function (PDF) for the Kumaraswamy-Kumaraswamy (kkw) distribution with parameters `alpha` ($\alpha$), `beta` ($\beta$), `delta` ($\delta$), and `lambda` ($\lambda$). This distribution is defined on the interval (0, 1). ```r dkkw(x, alpha, beta, delta, 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. - `delta`: Shape parameter `delta` >= 0. Can be a scalar or a vector. Default: 0.0. - `lambda`: Shape parameter `lambda` > 0. 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`, `delta`, `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`, `delta < 0`, `lambda <= 0`). ## Details The Kumaraswamy-Kumaraswamy (kkw) distribution is a special case of the five-parameter Generalized Kumaraswamy distribution (`dgkw`) obtained by setting the parameter $\gamma = 1$. The probability density function is given by: $$ f(x; \alpha, \beta, \delta, \lambda) = (\delta + 1) \lambda \alpha \beta x^{\alpha - 1} (1 - x^\alpha)^{\beta - 1} \bigl[1 - (1 - x^\alpha)^\beta\bigr]^{\lambda - 1} \bigl\{1 - \bigl[1 - (1 - x^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta} $$ for $0 < x < 1$. Note that $1/(\delta+1)$ corresponds to the Beta function term $B(1, \delta+1)$ when $\gamma=1$. Numerical evaluation follows similar stability considerations as `dgkw`. ## Examples ```r # Example values x_vals <- c(0.2, 0.5, 0.8) alpha_par <- 2.0 beta_par <- 3.0 delta_par <- 0.5 lambda_par <- 1.5 # Calculate density densities <- dkkw(x_vals, alpha_par, beta_par, delta_par, lambda_par) print(densities) # Calculate log-density log_densities <- dkkw(x_vals, alpha_par, beta_par, delta_par, lambda_par, log_prob = TRUE) print(log_densities) # Check: should match log(densities) print(log(densities)) # Compare with dgkw setting gamma = 1 densities_gkw <- dgkw(x_vals, alpha_par, beta_par, gamma = 1.0, delta_par, lambda_par) print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero # Plot the density curve_x <- seq(0.01, 0.99, length.out = 200) curve_y <- dkkw(curve_x, alpha_par, beta_par, delta_par, lambda_par) plot(curve_x, curve_y, type = "l", main = "kkw Density Example", xlab = "x", ylab = "f(x)", col = "blue") ``` ## References 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), `pkkw`, `qkkw`, `rkkw` (if they exist), `dbeta` ## Author(s) Lopes, J. E.

dkkw function