# Density of the Beta Distribution (gamma, delta+1 Parameterization) Computes the probability density function (PDF) for the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by `gamma` ($\gamma$) and `delta` ($\delta$), corresponding to the standard Beta distribution with shape parameters `shape1 = gamma` and `shape2 = delta + 1`. The distribution is defined on the interval (0, 1). ```r dbeta_(x, gamma, delta, log_prob = FALSE) ``` ## Arguments - `x`: Vector of quantiles (values between 0 and 1). - `gamma`: First shape parameter (`shape1`), $\gamma > 0$. Can be a scalar or a vector. Default: 1.0. - `delta`: Second shape parameter is `delta + 1` (`shape2`), requires $\delta \ge 0$ so that `shape2 >= 1`. Can be a scalar or a vector. Default: 0.0 (leading to `shape2 = 1`). - `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`). 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`). ## Details The probability density function (PDF) calculated by this function corresponds to a standard Beta distribution $Beta(\gamma, \delta+1)$: $$ f(x; \gamma, \delta) = \frac{x^{\gamma-1} (1-x)^{(\delta+1)-1}}{B(\gamma, \delta+1)} = \frac{x^{\gamma-1} (1-x)^{\delta}}{B(\gamma, \delta+1)} $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (`beta`). This specific parameterization arises as a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (`dgkw`) obtained by setting the parameters $\alpha = 1$, $\beta = 1$, and $\lambda = 1$. It is therefore equivalent to the McDonald (Mc)/Beta Power distribution (`dmc`) with $\lambda = 1$. Note the difference in the second parameter compared to `dbeta`, where `dbeta(x, shape1, shape2)` uses `shape2` directly. Here, `shape1 = gamma` and `shape2 = delta + 1`. ## Examples ```r # Example values x_vals <- c(0.2, 0.5, 0.8) gamma_par <- 2.0 # Corresponds to shape1 delta_par <- 3.0 # Corresponds to shape2 - 1 shape1 <- gamma_par shape2 <- delta_par + 1 # Calculate density using dbeta_ densities <- dbeta_(x_vals, gamma_par, delta_par) print(densities) # Compare with stats::dbeta densities_stats <- stats::dbeta(x_vals, shape1 = shape1, shape2 = shape2) print(paste("Max difference vs stats::dbeta:", max(abs(densities - densities_stats)))) # Compare with dgkw setting alpha=1, beta=1, lambda=1 densities_gkw <- dgkw(x_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par, delta = delta_par, lambda = 1.0) print(paste("Max difference vs dgkw:", max(abs(densities - densities_gkw)))) # Compare with dmc setting lambda=1 densities_mc <- dmc(x_vals, gamma = gamma_par, delta = delta_par, lambda = 1.0) print(paste("Max difference vs dmc:", max(abs(densities - densities_mc)))) # Calculate log-density log_densities <- dbeta_(x_vals, gamma_par, delta_par, log_prob = TRUE) print(log_densities) print(stats::dbeta(x_vals, shape1 = shape1, shape2 = shape2, log = TRUE)) # Plot the density curve_x <- seq(0.001, 0.999, length.out = 200) curve_y <- dbeta_(curve_x, gamma = 2, delta = 3) # Beta(2, 4) plot(curve_x, curve_y, type = "l", main = "Beta(2, 4) Density via dbeta_", xlab = "x", ylab = "f(x)", col = "blue") curve(stats::dbeta(x, 2, 4), add=TRUE, col="red", lty=2) legend("topright", legend=c("dbeta_(gamma=2, delta=3)", "stats::dbeta(shape1=2, shape2=4)"), col=c("blue", "red"), lty=c(1,2), bty="n") ``` ## References Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). **Continuous Univariate Distributions, Volume 2** (2nd ed.). Wiley. Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. **Journal of Statistical Computation and Simulation**, ## See Also `dbeta` (standard R implementation), `dgkw` (parent distribution density), `dmc` (McDonald/Beta Power density), `pbeta_`, `qbeta_`, `rbeta_` (other functions for this parameterization, if they exist). ## Author(s) Lopes, J. E.

dbeta_ function