Cumulative Distribution Function (CDF) of the EKw Distribution

Computes the cumulative distribution function (CDF), $P(X \le q)$, for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), and lambda ($\lambda$). This distribution is defined on the interval (0, 1) and is a special case of the Generalized Kumaraswamy (GKw) distribution where $\gamma = 1$ and $\delta = 0$.

Usage

pekw(q, alpha = 1, beta = 1, 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.
lambda: Shape parameter lambda > 0 (exponent parameter). 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, 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 Exponentiated Kumaraswamy (EKw) distribution is a special case of the five-parameter Generalized Kumaraswamy distribution (pgkw) obtained by setting parameters $\gamma = 1$ and $\delta = 0$.

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 $\gamma=1$ and $\delta=0$ gives $I_{y(q)}(1, 1)$. Since $I_x(1, 1) = x$, the CDF simplifies to $y(q)$: $$ F(q; \alpha, \beta, \lambda) = \bigl[1 - (1 - q^\alpha)^\beta \bigr]^\lambda $$ for $0 < q < 1$. The implementation uses this closed-form expression for efficiency and handles lower.tail and log.p arguments appropriately.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

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)
pekw(q, alpha = 2, beta = 3, lambda = 1.2)
#> [1] 0.07482254 0.51811492 0.94427733
pekw(q, alpha = 2, beta = 3, lambda = 1.2, lower.tail = FALSE)  # P(X > q)
#> [1] 0.92517746 0.48188508 0.05572267
pekw(q, alpha = 2, beta = 3, lambda = 1.2, log.p = TRUE)
#> [1] -2.59263616 -0.65755820 -0.05733537

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