# Density of the Kumaraswamy (Kw) Distribution Computes the probability density function (PDF) for the two-parameter Kumaraswamy (Kw) distribution with shape parameters `alpha` ($\alpha$) and `beta` ($\beta$). This distribution is defined on the interval (0, 1). ```r dkw(x, alpha, beta, 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. - `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`). 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`). ## Details The probability density function (PDF) of the Kumaraswamy (Kw) distribution is given by: $$ f(x; \alpha, \beta) = \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} $$ for $0 < x < 1$, $\alpha > 0$, and $\beta > 0$. The Kumaraswamy distribution is identical to the Generalized Kumaraswamy (GKw) distribution (`dgkw`) with parameters $\gamma = 1$, $\delta = 0$, and $\lambda = 1$. It is also a special case of the Exponentiated Kumaraswamy (`dekw`) with $\lambda = 1$, and the Kumaraswamy-Kumaraswamy (`dkkw`) with $\delta = 0$ and $\lambda = 1$. ## Examples ```r # Example values x_vals <- c(0.2, 0.5, 0.8) alpha_par <- 2.0 beta_par <- 3.0 # Calculate density using dkw densities <- dkw(x_vals, alpha_par, beta_par) print(densities) # Calculate log-density log_densities <- dkw(x_vals, alpha_par, beta_par, log_prob = TRUE) print(log_densities) # Check: should match log(densities) print(log(densities)) # Compare with dgkw setting gamma = 1, delta = 0, lambda = 1 densities_gkw <- dgkw(x_vals, alpha_par, beta_par, gamma = 1.0, delta = 0.0, lambda = 1.0) print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero # Plot the density for different shape parameter combinations curve_x <- seq(0.001, 0.999, length.out = 200) plot(curve_x, dkw(curve_x, alpha = 2, beta = 3), type = "l", main = "Kumaraswamy Density Examples", xlab = "x", ylab = "f(x)", col = "blue", ylim = c(0, 4)) lines(curve_x, dkw(curve_x, alpha = 3, beta = 2), col = "red") lines(curve_x, dkw(curve_x, alpha = 0.5, beta = 0.5), col = "green") # U-shaped lines(curve_x, dkw(curve_x, alpha = 5, beta = 1), col = "purple") # J-shaped lines(curve_x, dkw(curve_x, alpha = 1, beta = 3), col = "orange") # J-shaped (reversed) legend("top", legend = c("a=2, b=3", "a=3, b=2", "a=0.5, b=0.5", "a=5, b=1", "a=1, b=3"), col = c("blue", "red", "green", "purple", "orange"), lty = 1, bty = "n", ncol = 2) ``` ## References Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. **Journal of Hydrology**, **46**(1-2), 79-88. Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. **Statistical Methodology**, **6**(1), 70-81. ## See Also `dgkw` (parent distribution density), `dekw`, `dkkw`, `pkw`, `qkw`, `rkw` (other Kw functions), `dbeta` ## Author(s) Lopes, J. E.

dkw function