Density of the McDonald (Mc)/Beta Power Distribution

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 = 1, delta = 0, lambda = 1, log = FALSE)

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: Logical; if TRUE, the logarithm of the density is returned ($\log(f(x))$). Default: 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 = TRUE) for x strictly outside the interval [0, 1]. At the closed boundaries x = 0 and x = 1 the limiting density is returned rather than 0, following the convention of base R's density functions (compare dbeta); depending on the parameters that limit is 0, a finite positive value, or Inf. 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.

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. 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

x <- c(0.1, 0.3, 0.5, 0.7, 0.9)
dmc(x, gamma = 0.5, delta = 5, lambda = 3)
#> [1] 1.277650206 1.939587413 1.472684770 0.415872685 0.005630568
dmc(x, gamma = 0.5, delta = 5, lambda = 3, log = TRUE)
#> [1]  0.2450226  0.6624753  0.3870871 -0.8773761 -5.1795449

## Mc is GKw with alpha = beta = 1
all.equal(dmc(x, 0.5, 5, 3), dgkw(x, 1, 1, 0.5, 5, 3))
#> [1] TRUE

## The density integrates to one
integrate(dmc, 0, 1, gamma = 0.5, delta = 5, lambda = 3, rel.tol = 1e-10)
#> 1 with absolute error < 1.4e-12

curve(dmc(x, gamma = 0.5, delta = 5, lambda = 3), from = 0, to = 1, ylab = "density")