# CDF of the McDonald (Mc)/Beta Power Distribution Computes the cumulative distribution function (CDF), $F(q) = P(X \le q)$, for the McDonald (Mc) distribution (also known as Beta Power) with parameters `gamma` ($\gamma$), `delta` ($\delta$), and `lambda` ($\lambda$). This distribution is defined on the interval (0, 1) and is a special case of the Generalized Kumaraswamy (GKw) distribution where $\alpha = 1$ and $\beta = 1$. ```r pmc(q, gamma, delta, lambda, lower_tail = TRUE, log_p = FALSE) ``` ## Arguments - `q`: Vector of quantiles (values generally between 0 and 1). - `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/complements depending on `lower_tail` and `log_p`. The length of the result is determined by the recycling rule applied to the arguments (`q`, `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 McDonald (Mc) distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (`pgkw`) obtained by setting parameters $\alpha = 1$ and $\beta = 1$. 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 $\alpha=1$ and $\beta=1$ simplifies $y(q)$ to $q^\lambda$, yielding the Mc CDF: $$ F(q; \gamma, \delta, \lambda) = I_{q^\lambda}(\gamma, \delta+1) $$ This is evaluated using the `pbeta` function as `pbeta(q^lambda, shape1 = gamma, shape2 = delta + 1)`. ## Examples ```r # Example values q_vals <- c(0.2, 0.5, 0.8) gamma_par <- 2.0 delta_par <- 1.5 lambda_par <- 1.0 # Equivalent to Beta(gamma, delta+1) # Calculate CDF P(X <= q) using pmc probs <- pmc(q_vals, gamma_par, delta_par, lambda_par) print(probs) # Compare with Beta CDF print(stats::pbeta(q_vals, shape1 = gamma_par, shape2 = delta_par + 1)) # Calculate upper tail P(X > q) probs_upper <- pmc(q_vals, gamma_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 <- pmc(q_vals, gamma_par, delta_par, lambda_par, log_p = TRUE) print(log_probs) # Check: should match log(probs) print(log(probs)) # Compare with pgkw setting alpha = 1, beta = 1 probs_gkw <- pgkw(q_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par, delta = delta_par, lambda = lambda_par) print(paste("Max difference:", max(abs(probs - probs_gkw)))) # Should be near zero # Plot the CDF for different lambda values curve_q <- seq(0.01, 0.99, length.out = 200) curve_p1 <- pmc(curve_q, gamma = 2, delta = 3, lambda = 0.5) curve_p2 <- pmc(curve_q, gamma = 2, delta = 3, lambda = 1.0) # Beta(2, 4) curve_p3 <- pmc(curve_q, gamma = 2, delta = 3, lambda = 2.0) plot(curve_q, curve_p2, type = "l", main = "Mc/Beta Power CDF (gamma=2, delta=3)", xlab = "q", ylab = "F(q)", col = "red", ylim = c(0, 1)) lines(curve_q, curve_p1, col = "blue") lines(curve_q, curve_p3, col = "green") legend("bottomright", legend = c("lambda=0.5", "lambda=1.0 (Beta)", "lambda=2.0"), col = c("blue", "red", "green"), lty = 1, bty = "n") ``` ## References McDonald, J. B. (1984). Some generalized functions for the size distribution of income. **Econometrica**, 52(3), 647-663. 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), `dmc`, `qmc`, `rmc` (other Mc functions), `pbeta` ## Author(s) Lopes, J. E.

pmc function