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 = 1, delta = 0, lambda = 1, 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).
An out-of-bound or missing parameter is an error, not a return value: the wrapper stops with a message naming the parameter. An infinite parameter is not currently intercepted there and reaches the C++ layer, which treats it as invalid.

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. doi:10.2307/1913469

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898. doi:10.1080/00949650903530745

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88. doi:10.1016/0022-1694(80)90036-0

Author

Lopes, J. E.

Examples

p <- c(0.1, 0.5, 0.9)
qmc(p, gamma = 0.5, delta = 5, lambda = 3)
#> [1] 0.1110876 0.3383841 0.5937371
# upper-tail quantiles
qmc(p, gamma = 0.5, delta = 5, lambda = 3, lower.tail = FALSE)
#> [1] 0.5937371 0.3383841 0.1110876

## qmc() inverts pmc()
all.equal(pmc(qmc(p, gamma = 0.5, delta = 5, lambda = 3), gamma = 0.5,
    delta = 5, lambda = 3), p)
#> [1] TRUE