# Generalized Kumaraswamy Distribution CDF Computes the cumulative distribution function (CDF) for the five-parameter Generalized Kumaraswamy (GKw) distribution, defined on the interval (0, 1). Calculates $P(X \le q)$. ```r pgkw(q, alpha, beta, gamma, delta, lambda, lower_tail = TRUE, log_p = FALSE) ``` ## Arguments - `q`: Vector of quantiles (values generally 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. - `lambda`: Shape parameter `lambda` > 0. Can be a scalar or a vector. Default: 1.0. - `lower_tail`: Logical; if `TRUE` (default), probabilities are $P(X \le q)$, otherwise, $P(X > q)$. - `log_p`: Logical; if `TRUE`, probabilities $p$ are given as $\log(p)$. Default: `FALSE`. ## Returns A vector of probabilities, $F(q)$, or their logarithms if `log_p = TRUE`. The length of the result is determined by the recycling rule applied to the arguments (`q`, `alpha`, `beta`, `gamma`, `delta`, `lambda`). Returns `0` (or `-Inf` if `log_p = TRUE`) for `q <= 0` and `1` (or `0` if `log_p = TRUE`) for `q >= 1`. Returns `NaN` for invalid parameters. ## Details The cumulative distribution function (CDF) of the Generalized Kumaraswamy (GKw) distribution with parameters `alpha` ($\alpha$), `beta` ($\beta$), `gamma` ($\gamma$), `delta` ($\delta$), and `lambda` ($\lambda$) is given by: $$ F(q; \alpha, \beta, \gamma, \delta, \lambda) =I_{x(q)}(\gamma, \delta+1) $$ where $x(q) = [1-(1-q^{\alpha})^{\beta}]^{\lambda}$ and $I_x(a, b)$ is the regularized incomplete beta function, defined as: $$ I_x(a, b) = \frac{B_x(a, b)}{B(a, b)} = \frac{\int_0^x t^{a-1}(1-t)^{b-1} dt}{\int_0^1 t^{a-1}(1-t)^{b-1} dt} $$ This corresponds to the `pbeta` function in R, such that $F(q; \alpha, \beta, \gamma, \delta, \lambda) = \code{pbeta}(x(q), \code{shape1} = \gamma, \code{shape2} = \delta+1)$. The GKw distribution includes several special cases, such as the Kumaraswamy, Beta, and Exponentiated Kumaraswamy distributions (see `dgkw` for details). The function utilizes numerical algorithms for computing the regularized incomplete beta function accurately, especially near the boundaries. ## Examples ```r # Simple CDF evaluation prob <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1) # Kw case print(prob) # Upper tail probability P(X > q) prob_upper <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1, lower_tail = FALSE) print(prob_upper) # Check: prob + prob_upper should be 1 print(prob + prob_upper) # Log probability log_prob <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1, log_p = TRUE) print(log_prob) # Check: exp(log_prob) should be prob print(exp(log_prob)) # Use of vectorized parameters q_vals <- c(0.2, 0.5, 0.8) alphas_vec <- c(0.5, 1.0, 2.0) betas_vec <- c(1.0, 2.0, 3.0) # Vectorizes over q, alpha, beta pgkw(q_vals, alpha = alphas_vec, beta = betas_vec, gamma = 1, delta = 0.5, lambda = 0.5) # Plotting the CDF for special cases x_seq <- seq(0.01, 0.99, by = 0.01) # Standard Kumaraswamy CDF cdf_kw <- pgkw(x_seq, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1) # Beta distribution CDF equivalent (Beta(gamma, delta+1)) cdf_beta_equiv <- pgkw(x_seq, alpha = 1, beta = 1, gamma = 2, delta = 3, lambda = 1) # Compare with stats::pbeta cdf_beta_check <- stats::pbeta(x_seq, shape1 = 2, shape2 = 3 + 1) # max(abs(cdf_beta_equiv - cdf_beta_check)) # Should be close to zero plot(x_seq, cdf_kw, type = "l", ylim = c(0, 1), main = "GKw CDF Examples", ylab = "F(x)", xlab = "x", col = "blue") lines(x_seq, cdf_beta_equiv, col = "red", lty = 2) legend("bottomright", legend = c("Kw(2,3)", "Beta(2,4) equivalent"), col = c("blue", "red"), lty = c(1, 2), 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`, `qgkw`, `rgkw`, `pbeta` ## Author(s) Lopes, J. E.

pgkw function