# Cumulative Distribution Function (CDF) of the kkw Distribution Computes the cumulative distribution function (CDF), $P(X \le q)$, 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 pkkw(q, alpha, beta, 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. - `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/complements depending on `lower_tail` and `log_p`. The length of the result is determined by the recycling rule applied to the arguments (`q`, `alpha`, `beta`, `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 Kumaraswamy-Kumaraswamy (kkw) distribution is a special case of the five-parameter Generalized Kumaraswamy distribution (`pgkw`) obtained by setting the shape parameter $\gamma = 1$. The CDF of the GKw distribution is $F_{GKw}(q) = I_{y(q)}(\gamma, \delta+1)$, where $y(q) = [1-(1-q^{\alpha})^{\beta}]^{\lambda}$ and $I_x(a,b)$ is the regularized incomplete beta function (`pbeta`). Setting $\gamma=1$ utilizes the property $I_x(1, b) = 1 - (1-x)^b$, yielding the kkw CDF: $$ F(q; \alpha, \beta, \delta, \lambda) = 1 - \bigl\{1 - \bigl[1 - (1 - q^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta + 1} $$ for $0 < q < 1$. The implementation uses this closed-form expression for efficiency and handles `lower_tail` and `log_p` arguments appropriately. ## Examples ```r # Example values q_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 CDF P(X <= q) probs <- pkkw(q_vals, alpha_par, beta_par, delta_par, lambda_par) print(probs) # Calculate upper tail P(X > q) probs_upper <- pkkw(q_vals, alpha_par, beta_par, delta_par, lambda_par, lower_tail = FALSE) print(probs_upper) # Check: probs + probs_upper should be 1 print(probs + probs_upper) # Calculate log CDF log_probs <- pkkw(q_vals, alpha_par, beta_par, delta_par, lambda_par, log_p = TRUE) print(log_probs) # Check: should match log(probs) print(log(probs)) # Compare with pgkw setting gamma = 1 probs_gkw <- pgkw(q_vals, alpha_par, beta_par, gamma = 1.0, delta_par, lambda_par) print(paste("Max difference:", max(abs(probs - probs_gkw)))) # Should be near zero # Plot the CDF curve_q <- seq(0.01, 0.99, length.out = 200) curve_p <- pkkw(curve_q, alpha_par, beta_par, delta_par, lambda_par) plot(curve_q, curve_p, type = "l", main = "kkw CDF Example", xlab = "q", ylab = "F(q)", col = "blue", ylim = c(0, 1)) ``` ## 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 `pgkw` (parent distribution CDF), `dkkw`, `qkkw`, `rkkw`, `pbeta` ## Author(s) Lopes, J. E.

pkkw function