Generalized Kumaraswamy Distribution CDF

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, beta, gamma, delta, lambda, 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 if log_p = TRUE. The length of the result is determined by the recycling rule applied to the arguments (q, alpha, beta, gamma, delta, lambda). Returns 0 (or -Inf if log_p = TRUE) for q <= 0 and 1 (or 0 if log_p = TRUE) for q >= 1. Returns NaN for invalid parameters.

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

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

Author

Lopes, J. E.

Examples

# \donttest{
# Simple CDF evaluation
prob <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1) # Kw case
print(prob)
#> [1] 0.578125

# Upper tail probability P(X > q)
prob_upper <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1,
                 lower_tail = FALSE)
print(prob_upper)
#> [1] 0.421875
# Check: prob + prob_upper should be 1
print(prob + prob_upper)
#> [1] 1

# Log probability
log_prob <- pgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1,
                 log_p = TRUE)
print(log_prob)
#> [1] -0.5479652
# Check: exp(log_prob) should be prob
print(exp(log_prob))
#> [1] 0.578125

# Use of vectorized parameters
q_vals <- c(0.2, 0.5, 0.8)
alphas_vec <- c(0.5, 1.0, 2.0)
betas_vec <- c(1.0, 2.0, 3.0)
# Vectorizes over q, alpha, beta
pgkw(q_vals, alpha = alphas_vec, beta = betas_vec, gamma = 1, delta = 0.5, lambda = 0.5)
#> [1] 0.8093429 0.9509619 0.9963730

# Plotting the CDF for special cases
x_seq <- seq(0.01, 0.99, by = 0.01)
# Standard Kumaraswamy CDF
cdf_kw <- pgkw(x_seq, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1)
# Beta distribution CDF equivalent (Beta(gamma, delta+1))
cdf_beta_equiv <- pgkw(x_seq, alpha = 1, beta = 1, gamma = 2, delta = 3, lambda = 1)
# Compare with stats::pbeta
cdf_beta_check <- stats::pbeta(x_seq, shape1 = 2, shape2 = 3 + 1)
# max(abs(cdf_beta_equiv - cdf_beta_check)) # Should be close to zero

plot(x_seq, cdf_kw, type = "l", ylim = c(0, 1),
     main = "GKw CDF Examples", ylab = "F(x)", xlab = "x", col = "blue")
lines(x_seq, cdf_beta_equiv, col = "red", lty = 2)
legend("bottomright", legend = c("Kw(2,3)", "Beta(2,4) equivalent"),
       col = c("blue", "red"), lty = c(1, 2), bty = "n")

# }