Quantile Function of the McDonald (Mc)/Beta Power Distribution

Computes the quantile function (inverse CDF) for the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma ($\gamma$), delta ($\delta$), and lambda ($\lambda$). It finds the value q such that $P(X \le q) = p$. This distribution is a special case of the Generalized Kumaraswamy (GKw) distribution where $\alpha = 1$ and $\beta = 1$.

Usage

qmc(p, gamma, delta, lambda, lower_tail = TRUE, log_p = FALSE)

Arguments

p: Vector of probabilities (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.
lower_tail: Logical; if TRUE (default), probabilities are $p = P(X \le q)$, otherwise, probabilities are $p = P(X > q)$.
log_p: Logical; if TRUE, probabilities p are given as $\log(p)$. Default: FALSE.

Value

A vector of quantiles corresponding to the given probabilities p. The length of the result is determined by the recycling rule applied to the arguments (p, gamma, delta, lambda). Returns:

0 for p = 0 (or p = -Inf if log_p = TRUE, when lower_tail = TRUE).
1 for p = 1 (or p = 0 if log_p = TRUE, when lower_tail = TRUE).
NaN for p < 0 or p > 1 (or corresponding log scale).
NaN for invalid parameters (e.g., gamma <= 0, delta < 0, lambda <= 0).

Boundary return values are adjusted accordingly for lower_tail = FALSE.

Details

The quantile function $Q(p)$ is the inverse of the CDF $F(q)$. The CDF for the Mc ($\alpha=1, \beta=1$) distribution is $F(q) = I_{q^\lambda}(\gamma, \delta+1)$, where $I_z(a,b)$ is the regularized incomplete beta function (see pmc).

To find the quantile $q$, we first invert the Beta function part: let $y = I^{-1}_{p}(\gamma, \delta+1)$, where $I^{-1}_p(a,b)$ is the inverse computed via qbeta. We then solve $q^\lambda = y$ for $q$, yielding the quantile function: $$ Q(p) = \left[ I^{-1}_{p}(\gamma, \delta+1) \right]^{1/\lambda} $$ The function uses this formula, calculating $I^{-1}_{p}(\gamma, \delta+1)$ via qbeta(p, gamma, delta + 1, ...) while respecting the lower_tail and log_p arguments. This is equivalent to the general GKw quantile function (qgkw) evaluated with $\alpha=1, \beta=1$.

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
p_vals <- c(0.1, 0.5, 0.9)
gamma_par <- 2.0
delta_par <- 1.5
lambda_par <- 1.0 # Equivalent to Beta(gamma, delta+1)

# Calculate quantiles using qmc
quantiles <- qmc(p_vals, gamma_par, delta_par, lambda_par)
print(quantiles)
#> [1] 0.1649288 0.4355544 0.7379563
# Compare with Beta quantiles
print(stats::qbeta(p_vals, shape1 = gamma_par, shape2 = delta_par + 1))
#> [1] 0.1649288 0.4355544 0.7379563

# Calculate quantiles for upper tail probabilities P(X > q) = p
quantiles_upper <- qmc(p_vals, gamma_par, delta_par, lambda_par,
                       lower_tail = FALSE)
print(quantiles_upper)
#> [1] 0.7379563 0.4355544 0.1649288
# Check: qmc(p, ..., lt=F) == qmc(1-p, ..., lt=T)
print(qmc(1 - p_vals, gamma_par, delta_par, lambda_par))
#> [1] 0.7379563 0.4355544 0.1649288

# Calculate quantiles from log probabilities
log_p_vals <- log(p_vals)
quantiles_logp <- qmc(log_p_vals, gamma_par, delta_par, lambda_par, log_p = TRUE)
print(quantiles_logp)
#> [1] 0.1649288 0.4355544 0.7379563
# Check: should match original quantiles
print(quantiles)
#> [1] 0.1649288 0.4355544 0.7379563

# Compare with qgkw setting alpha = 1, beta = 1
quantiles_gkw <- qgkw(p_vals, alpha = 1.0, beta = 1.0, gamma = gamma_par,
                      delta = delta_par, lambda = lambda_par)
print(paste("Max difference:", max(abs(quantiles - quantiles_gkw)))) # Should be near zero
#> [1] "Max difference: 5.55111512312578e-17"

# Verify inverse relationship with pmc
p_check <- 0.75
q_calc <- qmc(p_check, gamma_par, delta_par, lambda_par) # Use lambda != 1
p_recalc <- pmc(q_calc, gamma_par, delta_par, lambda_par)
print(paste("Original p:", p_check, " Recalculated p:", p_recalc))
#> [1] "Original p: 0.75  Recalculated p: 0.75"
# abs(p_check - p_recalc) < 1e-9 # Should be TRUE

# Boundary conditions
print(qmc(c(0, 1), gamma_par, delta_par, lambda_par)) # Should be 0, 1
#> [1] 0 1
print(qmc(c(-Inf, 0), gamma_par, delta_par, lambda_par, log_p = TRUE)) # Should be 0, 1
#> [1] 0 1

# }