Cumulative Distribution Function (CDF) of the Kumaraswamy (Kw) Distribution

Computes the cumulative distribution function (CDF), $P(X \le q)$, 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

pkw(q, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

q: Vector of quantiles (values generally 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.
lower.tail: Logical; if TRUE (default), probabilities are $P(X \le q)$, otherwise, $P(X > q)$.
log.p: Logical; if TRUE, probabilities $p$ are given as $\log(p)$. Default: FALSE.

Value

A vector of probabilities, $F(q)$, or their logarithms/complements depending on lower.tail and log.p. The length of the result is determined by the recycling rule applied to the arguments (q, alpha, beta). When lower.tail = TRUE, returns 0 (or -Inf if log.p = TRUE) for q <= 0 and 1 (or 0 if log.p = TRUE) for q >= 1. 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. Boundary return values are adjusted accordingly for lower.tail = FALSE.

Details

The cumulative distribution function (CDF) of the Kumaraswamy (Kw) distribution is given by: $$ F(x; \alpha, \beta) = 1 - (1 - x^\alpha)^\beta $$ for $0 < x < 1$, $\alpha > 0$, and $\beta > 0$.

The Kw distribution is a special case of several generalized distributions:

Generalized Kumaraswamy (pgkw) with $\gamma=1, \delta=0, \lambda=1$.
Exponentiated Kumaraswamy (pekw) with $\lambda=1$.
Kumaraswamy-Kumaraswamy (pkkw) with $\delta=0, \lambda=1$.

The implementation uses the closed-form expression for efficiency.

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

q <- c(0.2, 0.5, 0.8)
pkw(q, alpha = 2, beta = 3)
#> [1] 0.115264 0.578125 0.953344
pkw(q, alpha = 2, beta = 3, lower.tail = FALSE)  # P(X > q)
#> [1] 0.884736 0.421875 0.046656
pkw(q, alpha = 2, beta = 3, log.p = TRUE)
#> [1] -2.16053013 -0.54796517 -0.04777948

## pkw() is the integral of dkw()
Fq <- pkw(0.5, alpha = 2, beta = 3)
all.equal(Fq, integrate(dkw, 0, 0.5, alpha = 2, beta = 3, rel.tol = 1e-10)$value)
#> [1] TRUE