CDF of the McDonald (Mc)/Beta Power Distribution

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.

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.

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).

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.

Author

Lopes, J. E.

Examples

# \donttest{
# 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)
#> [1] 0.1413499 0.6022524 0.9463344
# Compare with Beta CDF
print(stats::pbeta(q_vals, shape1 = gamma_par, shape2 = delta_par + 1))
#> [1] 0.1413499 0.6022524 0.9463344

# Calculate upper tail P(X > q)
probs_upper <- pmc(q_vals, gamma_par, delta_par, lambda_par,
                   lower_tail = FALSE)
print(probs_upper)
#> [1] 0.85865010 0.39774756 0.05366563
# Check: probs + probs_upper should be 1
print(probs + probs_upper)
#> [1] 1 1 1

# Calculate log CDF
log_probs <- pmc(q_vals, gamma_par, delta_par, lambda_par, log_p = TRUE)
print(log_probs)
#> [1] -1.95651693 -0.50707859 -0.05515932
# Check: should match log(probs)
print(log(probs))
#> [1] -1.95651693 -0.50707859 -0.05515932

# 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
#> [1] "Max difference: 5.55111512312578e-17"

# 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")

# }