gkwreg (version 1.0.7)

dmc: Density of the McDonald (Mc)/Beta Power Distribution Distribution

Description

Computes the probability density function (PDF) for the McDonald (Mc) distribution (also previously referred to as Beta Power) with parameters gamma (\(\gamma\)), delta (\(\delta\)), and lambda (\(\lambda\)). This distribution is defined on the interval (0, 1).

Usage

dmc(x, gamma, delta, lambda, log_prob = FALSE)

Value

A vector of density values (\(f(x)\)) or log-density values (\(\log(f(x))\)). The length of the result is determined by the recycling rule applied to the arguments (x, gamma, delta, lambda). Returns 0 (or -Inf if log_prob = TRUE) for x outside the interval (0, 1), or NaN if parameters are invalid (e.g., gamma <= 0, delta < 0, lambda <= 0).

Arguments

x

Vector of quantiles (values 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.

log_prob

Logical; if TRUE, the logarithm of the density is returned (\(\log(f(x))\)). Default: FALSE.

Author

Lopes, J. E.

Details

The probability density function (PDF) of the McDonald (Mc) distribution is given by: $$ f(x; \gamma, \delta, \lambda) = \frac{\lambda}{B(\gamma,\delta+1)} x^{\gamma \lambda - 1} (1 - x^\lambda)^\delta $$ for \(0 < x < 1\), where \(B(a,b)\) is the Beta function (beta).

The Mc distribution is a special case of the five-parameter Generalized Kumaraswamy (GKw) distribution (dgkw) obtained by setting the parameters \(\alpha = 1\) and \(\beta = 1\). It was introduced by McDonald (1984) and is related to the Generalized Beta distribution of the first kind (GB1). When \(\lambda=1\), it simplifies to the standard Beta distribution with parameters \(\gamma\) and \(\delta+1\).

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.

See Also

dgkw (parent distribution density), pmc, qmc, rmc (other Mc functions), dbeta

Examples

# \donttest{
# Example values
x_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 density using dmc
densities <- dmc(x_vals, gamma_par, delta_par, lambda_par)
print(densities)
# Compare with Beta density
print(stats::dbeta(x_vals, shape1 = gamma_par, shape2 = delta_par + 1))

# Calculate log-density
log_densities <- dmc(x_vals, gamma_par, delta_par, lambda_par, log_prob = TRUE)
print(log_densities)

# Compare with dgkw setting alpha = 1, beta = 1
densities_gkw <- dgkw(x_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par,
                      delta = delta_par, lambda = lambda_par)
print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero

# Plot the density for different lambda values
curve_x <- seq(0.01, 0.99, length.out = 200)
curve_y1 <- dmc(curve_x, gamma = 2, delta = 3, lambda = 0.5)
curve_y2 <- dmc(curve_x, gamma = 2, delta = 3, lambda = 1.0) # Beta(2, 4)
curve_y3 <- dmc(curve_x, gamma = 2, delta = 3, lambda = 2.0)

plot(curve_x, curve_y2, type = "l", main = "McDonald (Mc) Density (gamma=2, delta=3)",
     xlab = "x", ylab = "f(x)", col = "red", ylim = range(0, curve_y1, curve_y2, curve_y3))
lines(curve_x, curve_y1, col = "blue")
lines(curve_x, curve_y3, col = "green")
legend("topright", legend = c("lambda=0.5", "lambda=1.0 (Beta)", "lambda=2.0"),
       col = c("blue", "red", "green"), lty = 1, bty = "n")
# }