Density of the Kumaraswamy (Kw) Distribution

Computes the probability density function (PDF) for the two-parameter Kumaraswamy (Kw) distribution with shape parameters alpha ($\alpha$) and beta ($\beta$). This distribution is defined on the interval (0, 1).

Usage

dkw(x, alpha = 1, beta = 1, 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.
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). 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 Kumaraswamy (Kw) distribution is given by: $$ f(x; \alpha, \beta) = \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} $$ for $0 < x < 1$, $\alpha > 0$, and $\beta > 0$.

The Kumaraswamy distribution is identical to the Generalized Kumaraswamy (GKw) distribution (dgkw) with parameters $\gamma = 1$, $\delta = 0$, and $\lambda = 1$. It is also a special case of the Exponentiated Kumaraswamy (dekw) with $\lambda = 1$, and the Kumaraswamy-Kumaraswamy (dkkw) with $\delta = 0$ and $\lambda = 1$.

References

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

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81. doi:10.1016/j.stamet.2008.04.001

Author

Lopes, J. E.

Examples

x <- c(0.1, 0.3, 0.5, 0.7, 0.9)
dkw(x, alpha = 2, beta = 3)
#> [1] 0.58806 1.49058 1.68750 1.09242 0.19494
dkw(x, alpha = 2, beta = 3, log = TRUE)
#> [1] -0.53092630  0.39916531  0.52324814  0.08839542 -1.63506346

## Closed form: alpha * beta * x^(alpha - 1) * (1 - x^alpha)^(beta - 1)
all.equal(dkw(x, 2, 3), 2 * 3 * x * (1 - x^2)^2)
#> [1] TRUE

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

curve(dkw(x, alpha = 2, beta = 3), from = 0, to = 1, ylab = "density")