Cumulative Distribution Function (CDF) of the KKw Distribution

Computes the cumulative distribution function (CDF), $P(X \le q)$, for the Kumaraswamy-Kumaraswamy (KKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), delta ($\delta$), and lambda ($\lambda$). This distribution is defined on the interval (0, 1).

Usage

pkkw(
  q,
  alpha = 1,
  beta = 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.
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, 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 Kumaraswamy-Kumaraswamy (KKw) distribution is a special case of the five-parameter Generalized Kumaraswamy distribution (pgkw) obtained by setting the shape parameter $\gamma = 1$.

The CDF of the GKw distribution is $F_{GKw}(q) = I_{y(q)}(\gamma, \delta+1)$, where $y(q) = [1-(1-q^{\alpha})^{\beta}]^{\lambda}$ and $I_x(a,b)$ is the regularized incomplete beta function (pbeta). Setting $\gamma=1$ utilizes the property $I_x(1, b) = 1 - (1-x)^b$, yielding the KKw CDF: $$ F(q; \alpha, \beta, \delta, \lambda) = 1 - \bigl\{1 - \bigl[1 - (1 - q^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta + 1} $$ for $0 < q < 1$.

The implementation uses this closed-form expression for efficiency and handles lower.tail and log.p arguments appropriately.

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

q <- c(0.2, 0.5, 0.8)
pkkw(q, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2)
#> [1] 0.1101075 0.6654853 0.9868463
# P(X > q)
pkkw(q, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2, lower.tail = FALSE)
#> [1] 0.8898925 0.3345147 0.0131537
pkkw(q, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2, log.p = TRUE)
#> [1] -2.20629852 -0.40723874 -0.01324097

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