gkwreg (version 1.0.7)

pkkw: Cumulative Distribution Function (CDF) of the kkw Distribution

Description

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, beta, delta, lambda, lower_tail = TRUE, log_p = 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). 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.

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.

Author

Lopes, J. E.

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

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), dkkw, qkkw, rkkw, pbeta

Examples

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

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

# Calculate upper tail P(X > q)
probs_upper <- pkkw(q_vals, alpha_par, beta_par, delta_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 <- pkkw(q_vals, alpha_par, beta_par, delta_par, lambda_par,
                   log_p = TRUE)
print(log_probs)
# Check: should match log(probs)
print(log(probs))

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

# Plot the CDF
curve_q <- seq(0.01, 0.99, length.out = 200)
curve_p <- pkkw(curve_q, alpha_par, beta_par, delta_par, lambda_par)
plot(curve_q, curve_p, type = "l", main = "kkw CDF Example",
     xlab = "q", ylab = "F(q)", col = "blue", ylim = c(0, 1))

# }