Density of the Beta-Kumaraswamy (BKw) Distribution

Computes the probability density function (PDF) for the Beta-Kumaraswamy (BKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), gamma ($\gamma$), and delta ($\delta$). This distribution is defined on the interval (0, 1).

Usage

dbkw(x, alpha = 1, beta = 1, gamma = 1, delta = 0, log = FALSE)

Arguments

x: Vector of quantiles (values 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.
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.
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, alpha, beta, 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, following the convention of base R's density functions (compare dbeta); 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) of the Beta-Kumaraswamy (BKw) distribution is given by: $$ f(x; \alpha, \beta, \gamma, \delta) = \frac{\alpha \beta}{B(\gamma, \delta+1)} x^{\alpha - 1} \bigl(1 - x^\alpha\bigr)^{\beta(\delta+1) - 1} \bigl[1 - \bigl(1 - x^\alpha\bigr)^\beta\bigr]^{\gamma - 1} $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (beta).

The BKw distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (dgkw) obtained by setting the parameter $\lambda = 1$. Numerical evaluation is performed using algorithms similar to those for dgkw, ensuring stability.

References

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

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88. doi:10.1016/0022-1694(80)90036-0

Author

Lopes, J. E.

Examples

x <- c(0.1, 0.3, 0.5, 0.7, 0.9)
dbkw(x, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)
#> [1] 0.25421436 1.63569904 2.12220174 0.94358703 0.04097103
dbkw(x, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, log = TRUE)
#> [1] -1.36957743  0.49207026  0.75245411 -0.05806668 -3.19488993

## BKw is GKw with lambda = 1
all.equal(dbkw(x, 2, 3, 1.5, 0.5), dgkw(x, 2, 3, 1.5, 0.5, lambda = 1))
#> [1] TRUE

## The density integrates to one
integrate(dbkw, 0, 1, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5,
    rel.tol = 1e-10)
#> 1 with absolute error < 2.5e-11

curve(dbkw(x, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5), from = 0,
    to = 1, ylab = "density")