pmc: CDF of the McDonald (Mc)/Beta Power Distribution

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

CDF of the McDonald (Mc)/Beta Power Distribution

Description

Computes the cumulative distribution function (CDF), F(q) = P(X \le q), for the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma (\gamma), delta (\delta), and lambda (\lambda). This distribution is defined on the interval (0, 1) and is a special case of the Generalized Kumaraswamy (GKw) distribution where \alpha = 1 and \beta = 1.

Usage

pmc(q, gamma, delta, lambda, lower_tail = TRUE, log_p = FALSE)

Arguments

q	Vector of quantiles (values generally between 0 and 1).
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.

Details

The McDonald (Mc) distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (pgkw) obtained by setting parameters \alpha = 1 and \beta = 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 \alpha=1 and \beta=1 simplifies y(q) to q^\lambda, yielding the Mc CDF:

F(q; \gamma, \delta, \lambda) = I_{q^\lambda}(\gamma, \delta+1)

This is evaluated using the pbeta function as pbeta(q^lambda, shape1 = gamma, shape2 = delta + 1).

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, 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.

Author(s)

Lopes, J. E.

References

McDonald, J. B. (1984). Some generalized functions for the size distribution of income. Econometrica, 52(3), 647-663.

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), dmc, qmc, rmc (other Mc functions), pbeta

Examples

# Example values
q_vals <- c(0.2, 0.5, 0.8)
gamma_par <- 2.0
delta_par <- 1.5
lambda_par <- 1.0 # Equivalent to Beta(gamma, delta+1)

# Calculate CDF P(X <= q) using pmc
probs <- pmc(q_vals, gamma_par, delta_par, lambda_par)
print(probs)
# Compare with Beta CDF
print(stats::pbeta(q_vals, shape1 = gamma_par, shape2 = delta_par + 1))

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

# Compare with pgkw setting alpha = 1, beta = 1
probs_gkw <- pgkw(q_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par,
                  delta = delta_par, 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 <- pmc(curve_q, gamma = 2, delta = 3, lambda = 0.5)
curve_p2 <- pmc(curve_q, gamma = 2, delta = 3, lambda = 1.0) # Beta(2, 4)
curve_p3 <- pmc(curve_q, gamma = 2, delta = 3, lambda = 2.0)

plot(curve_q, curve_p2, type = "l", main = "Mc/Beta Power CDF (gamma=2, delta=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 (Beta)", "lambda=2.0"),
       col = c("blue", "red", "green"), lty = 1, bty = "n")

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