# Density of the McDonald (Mc)/Beta Power Distribution Distribution Computes the probability density function (PDF) for the McDonald (Mc) distribution (also previously referred to as Beta Power) with parameters `gamma` ($\gamma$), `delta` ($\delta$), and `lambda` ($\lambda$). This distribution is defined on the interval (0, 1). ```r dmc(x, gamma, delta, lambda, log_prob = FALSE) ``` ## Arguments - `x`: Vector of quantiles (values 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. - `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`, `gamma`, `delta`, `lambda`). Returns `0` (or `-Inf` if `log_prob = TRUE`) for `x` outside the interval (0, 1), or `NaN` if parameters are invalid (e.g., `gamma <= 0`, `delta < 0`, `lambda <= 0`). ## Details The probability density function (PDF) of the McDonald (Mc) distribution is given by: $$ f(x; \gamma, \delta, \lambda) = \frac{\lambda}{B(\gamma,\delta+1)} x^{\gamma \lambda - 1} (1 - x^\lambda)^\delta $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (`beta`). The Mc distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (`dgkw`) obtained by setting the parameters $\alpha = 1$ and $\beta = 1$. It was introduced by McDonald (1984) and is related to the Generalized Beta distribution of the first kind (GB1). When $\lambda=1$, it simplifies to the standard Beta distribution with parameters $\gamma$ and $\delta+1$. ## Examples ```r # Example values x_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 density using dmc densities <- dmc(x_vals, gamma_par, delta_par, lambda_par) print(densities) # Compare with Beta density print(stats::dbeta(x_vals, shape1 = gamma_par, shape2 = delta_par + 1)) # Calculate log-density log_densities <- dmc(x_vals, gamma_par, delta_par, lambda_par, log_prob = TRUE) print(log_densities) # Compare with dgkw setting alpha = 1, beta = 1 densities_gkw <- dgkw(x_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par, delta = delta_par, lambda = lambda_par) print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero # Plot the density for different lambda values curve_x <- seq(0.01, 0.99, length.out = 200) curve_y1 <- dmc(curve_x, gamma = 2, delta = 3, lambda = 0.5) curve_y2 <- dmc(curve_x, gamma = 2, delta = 3, lambda = 1.0) # Beta(2, 4) curve_y3 <- dmc(curve_x, gamma = 2, delta = 3, lambda = 2.0) plot(curve_x, curve_y2, type = "l", main = "McDonald (Mc) Density (gamma=2, delta=3)", xlab = "x", ylab = "f(x)", col = "red", ylim = range(0, curve_y1, curve_y2, curve_y3)) lines(curve_x, curve_y1, col = "blue") lines(curve_x, curve_y3, col = "green") legend("topright", 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 `dgkw` (parent distribution density), `pmc`, `qmc`, `rmc` (other Mc functions), `dbeta` ## Author(s) Lopes, J. E.

dmc function