Cumulative Distribution Function (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 = 1, delta = 0, lambda = 1, 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). When lower.tail = TRUE, returns 0 (or -Inf if log.p = TRUE) for q <= 0 and 1 (or 0 if log.p = TRUE) for q >= 1. 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 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. 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

q <- c(0.2, 0.5, 0.8)
pmc(q, gamma = 0.5, delta = 5, lambda = 3)
#> [1] 0.2389267 0.7850539 0.9959906
pmc(q, gamma = 0.5, delta = 5, lambda = 3, lower.tail = FALSE)  # P(X > q)
#> [1] 0.761073272 0.214946121 0.004009438
pmc(q, gamma = 0.5, delta = 5, lambda = 3, log.p = TRUE)
#> [1] -1.431598352 -0.242002927 -0.004017497

## pmc() is the integral of dmc()
Fq <- pmc(0.5, gamma = 0.5, delta = 5, lambda = 3)
all.equal(Fq, integrate(dmc, 0, 0.5, gamma = 0.5, delta = 5, lambda = 3,
    rel.tol = 1e-10)$value)
#> [1] TRUE