Cumulative Distribution Function (CDF) of the Generalized Kumaraswamy Distribution

Computes the cumulative distribution function (CDF) for the five-parameter Generalized Kumaraswamy (GKw) distribution, defined on the interval (0, 1). Calculates $P(X \le q)$.

Usage

pgkw(
  q,
  alpha = 1,
  beta = 1,
  gamma = 1,
  delta = 0,
  lambda = 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.
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.
lambda: Shape parameter lambda > 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, gamma, delta, lambda). 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 Generalized Kumaraswamy (GKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), gamma ($\gamma$), delta ($\delta$), and lambda ($\lambda$) is given by: $$ F(q; \alpha, \beta, \gamma, \delta, \lambda) = I_{x(q)}(\gamma, \delta+1) $$ where $x(q) = [1-(1-q^{\alpha})^{\beta}]^{\lambda}$ and $I_x(a, b)$ is the regularized incomplete beta function, defined as: $$ I_x(a, b) = \frac{B_x(a, b)}{B(a, b)} = \frac{\int_0^x t^{a-1}(1-t)^{b-1} dt}{\int_0^1 t^{a-1}(1-t)^{b-1} dt} $$ This corresponds to the pbeta function in R, such that $F(q; \alpha, \beta, \gamma, \delta, \lambda) = \code{pbeta}(x(q), \code{shape1} = \gamma, \code{shape2} = \delta+1)$.

The GKw distribution includes several special cases, such as the Kumaraswamy, Beta, and Exponentiated Kumaraswamy distributions (see dgkw for details). The function utilizes numerical algorithms for computing the regularized incomplete beta function accurately, especially near the boundaries.

References

Carrasco, J. M. F., Ferrari, S. L. P., & Cordeiro, G. M. (2010). A new generalized Kumaraswamy distribution. arXiv preprint arXiv:1004.0911. doi:10.48550/arXiv.1004.0911

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

q <- c(0.2, 0.5, 0.8)
pgkw(q, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)
#> [1] 0.03395476 0.52305959 0.97804668
pgkw(q, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2,
    lower.tail = FALSE)  # P(X > q)
#> [1] 0.96604524 0.47694041 0.02195332
pgkw(q, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2,
    log.p = TRUE)
#> [1] -3.38272635 -0.64805989 -0.02219788

## pgkw() is the integral of dgkw()
Fq <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)
all.equal(Fq, integrate(dgkw, 0, 0.5, alpha = 2, beta = 3, gamma = 1.5,
    delta = 0.5, lambda = 1.2, rel.tol = 1e-10)$value)
#> [1] TRUE