pkkw: Cumulative Distribution Function (CDF) of the kkw...
| pkkw | R Documentation |
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)
Arguments
q |
Vector of quantiles (values generally between 0 and 1). |
alpha |
Shape parameter |
beta |
Shape parameter |
delta |
Shape parameter |
lambda |
Shape parameter |
lower_tail |
Logical; if |
log_p |
Logical; if |
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.
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.
Author(s)
Lopes, J. E.
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
# 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))
