pekw: Cumulative Distribution Function (CDF) of the EKw...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Cumulative Distribution Function (CDF) of the EKw Distribution

Description

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, beta, 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.
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.

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.

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). 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.

Author(s)

Lopes, J. E.

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,

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

See Also

pgkw (parent distribution CDF), dekw, qekw, rekw (other EKw functions),

Examples

# Example values
q_vals <- c(0.2, 0.5, 0.8)
alpha_par <- 2.0
beta_par <- 3.0
lambda_par <- 1.5

# Calculate CDF P(X <= q)
probs <- pekw(q_vals, alpha_par, beta_par, lambda_par)
print(probs)

# Calculate upper tail P(X > q)
probs_upper <- pekw(q_vals, alpha_par, beta_par, lambda_par,
                    lower_tail = FALSE)
print(probs_upper)
# Check: probs + probs_upper should be 1
print(probs + probs_upper)

# Calculate log CDF
log_probs <- pekw(q_vals, alpha_par, beta_par, lambda_par, log_p = TRUE)
print(log_probs)
# Check: should match log(probs)
print(log(probs))

# Compare with pgkw setting gamma = 1, delta = 0
probs_gkw <- pgkw(q_vals, alpha_par, beta_par, gamma = 1.0, delta = 0.0,
                 lambda = lambda_par)
print(paste("Max difference:", max(abs(probs - probs_gkw)))) # Should be near zero

# Plot the CDF for different lambda values
curve_q <- seq(0.01, 0.99, length.out = 200)
curve_p1 <- pekw(curve_q, alpha = 2, beta = 3, lambda = 0.5)
curve_p2 <- pekw(curve_q, alpha = 2, beta = 3, lambda = 1.0) # standard Kw
curve_p3 <- pekw(curve_q, alpha = 2, beta = 3, lambda = 2.0)

plot(curve_q, curve_p2, type = "l", main = "EKw CDF Examples (alpha=2, beta=3)",
     xlab = "q", ylab = "F(q)", col = "red", ylim = c(0, 1))
lines(curve_q, curve_p1, col = "blue")
lines(curve_q, curve_p3, col = "green")
legend("bottomright", legend = c("lambda=0.5", "lambda=1.0 (Kw)", "lambda=2.0"),
       col = c("blue", "red", "green"), lty = 1, bty = "n")

gkwreg documentation built on April 16, 2025, 1:10 a.m.