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).

Usage

dbeta_(x, gamma = 1, delta = 0, log = 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: Logical; if TRUE, the logarithm of the density is returned ($\log(f(x))$). Default: FALSE.

Value

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 = TRUE) for x strictly outside the interval [0, 1]. At the closed boundaries x = 0 and x = 1 the limiting density is returned rather than 0, matching dbeta with shape1 = gamma and shape2 = delta + 1; depending on the parameters that limit is 0, a finite positive value, or Inf. An out-of-bound or missing parameter is an error, not a return value: the wrapper stops with a message naming the parameter. An infinite parameter is not currently intercepted there and reaches the C++ layer, which treats it as invalid.

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.

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, 81(7), 883-898. doi:10.1080/00949650903530745

Author

Lopes, J. E.

Examples

x <- c(0.1, 0.3, 0.5, 0.7, 0.9)
dbeta_(x, gamma = 2, delta = 3)
#> [1] 1.458 2.058 1.250 0.378 0.018
dbeta_(x, gamma = 2, delta = 3, log = TRUE)
#> [1]  0.3770656  0.7217346  0.2231436 -0.9728611 -4.0173835

## The package's Beta(gamma, delta) is stats::dbeta with shapes gamma, delta + 1
all.equal(dbeta_(x, 2, 3), stats::dbeta(x, 2, 4))
#> [1] TRUE

## The density integrates to one
integrate(dbeta_, 0, 1, gamma = 2, delta = 3, rel.tol = 1e-10)
#> 1 with absolute error < 1.1e-14

curve(dbeta_(x, gamma = 2, delta = 3), from = 0, to = 1, ylab = "density")