# Density of the Beta-Kumaraswamy (BKw) Distribution Computes the probability density function (PDF) for the Beta-Kumaraswamy (BKw) distribution with parameters `alpha` ($\alpha$), `beta` ($\beta$), `gamma` ($\gamma$), and `delta` ($\delta$). This distribution is defined on the interval (0, 1). ```r dbkw(x, alpha, beta, gamma, delta, 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. - `gamma`: Shape parameter `gamma` > 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. - `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`, `gamma`, `delta`). 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`, `gamma <= 0`, `delta < 0`). ## Details The probability density function (PDF) of the Beta-Kumaraswamy (BKw) distribution is given by: $$ f(x; \alpha, \beta, \gamma, \delta) = \frac{\alpha \beta}{B(\gamma, \delta+1)} x^{\alpha - 1} \bigl(1 - x^\alpha\bigr)^{\beta(\delta+1) - 1} \bigl[1 - \bigl(1 - x^\alpha\bigr)^\beta\bigr]^{\gamma - 1} $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (`beta`). The BKw distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (`dgkw`) obtained by setting the parameter $\lambda = 1$. Numerical evaluation is performed using algorithms similar to those for `dgkw`, ensuring stability. ## Examples ```r # Example values x_vals <- c(0.2, 0.5, 0.8) alpha_par <- 2.0 beta_par <- 1.5 gamma_par <- 1.0 # Equivalent to Kw when gamma=1 delta_par <- 0.5 # Calculate density densities <- dbkw(x_vals, alpha_par, beta_par, gamma_par, delta_par) print(densities) # Calculate log-density log_densities <- dbkw(x_vals, alpha_par, beta_par, gamma_par, delta_par, log_prob = TRUE) print(log_densities) # Check: should match log(densities) print(log(densities)) # Compare with dgkw setting lambda = 1 densities_gkw <- dgkw(x_vals, alpha_par, beta_par, gamma = gamma_par, delta = delta_par, lambda = 1.0) print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero # Plot the density for different gamma values curve_x <- seq(0.01, 0.99, length.out = 200) curve_y1 <- dbkw(curve_x, alpha = 2, beta = 3, gamma = 0.5, delta = 1) curve_y2 <- dbkw(curve_x, alpha = 2, beta = 3, gamma = 1.0, delta = 1) curve_y3 <- dbkw(curve_x, alpha = 2, beta = 3, gamma = 2.0, delta = 1) plot(curve_x, curve_y1, type = "l", main = "BKw Density Examples (alpha=2, beta=3, delta=1)", xlab = "x", ylab = "f(x)", col = "blue", ylim = range(0, curve_y1, curve_y2, curve_y3)) lines(curve_x, curve_y2, col = "red") lines(curve_x, curve_y3, col = "green") legend("topright", legend = c("gamma=0.5", "gamma=1.0", "gamma=2.0"), col = c("blue", "red", "green"), lty = 1, bty = "n") ``` ## 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), `pbkw`, `qbkw`, `rbkw` (other BKw functions), ## Author(s) Lopes, J. E.

dbkw function