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Random Number Generation for the McDonald (Mc)/Beta Power Distribution

Source: R/bpmc.R
rmc.Rd

Generates random deviates from the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma (\(\gamma\)), delta (\(\delta\)), and lambda (\(\lambda\)). This distribution is a special case of the Generalized Kumaraswamy (GKw) distribution where \(\alpha = 1\) and \(\beta = 1\).

Usage

rmc(n, gamma = 1, delta = 0, lambda = 1)

Arguments

n

Number of observations. If length(n) > 1, the length is taken to be the number required. Must be a non-negative integer.

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.

Value

A vector of length n containing random deviates from the Mc distribution, with values in (0, 1). The length of the result is determined by n and the recycling rule applied to the parameters (gamma, delta, lambda). 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 generation method uses the relationship between the GKw distribution and the Beta distribution. The general procedure for GKw (rgkw) is: If \(W \sim \mathrm{Beta}(\gamma, \delta+1)\), then \(X = \{1 - [1 - W^{1/\lambda}]^{1/\beta}\}^{1/\alpha}\) follows the GKw(\(\alpha, \beta, \gamma, \delta, \lambda\)) distribution.

For the Mc distribution, \(\alpha=1\) and \(\beta=1\). Therefore, the algorithm simplifies significantly:

  1. Generate \(U \sim \mathrm{Beta}(\gamma, \delta+1)\) using rbeta.

  2. Compute the Mc variate \(X = U^{1/\lambda}\).

This procedure is implemented efficiently, handling parameter recycling as needed.

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

Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag. (General methods for random variate generation).

See also

rgkw (parent distribution random generation), dmc, pmc, qmc (other Mc functions), rbeta

Other random generation functions: rbeta_(), rbkw(), rekw(), rgkw(), rkkw(), rkw()

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rmc(1000, gamma = 0.5, delta = 5, lambda = 3)
summary(x)
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> 0.002166 0.189371 0.327717 0.334765 0.464918 0.873018 

## The sample follows the distribution
hist(x, breaks = 30, freq = FALSE, main = "")
curve(dmc(x, gamma = 0.5, delta = 5, lambda = 3), add = TRUE)

ks.test(x, pmc, gamma = 0.5, delta = 5, lambda = 3)
#> 
#> 	Asymptotic one-sample Kolmogorov-Smirnov test
#> 
#> data:  x
#> D = 0.035509, p-value = 0.1606
#> alternative hypothesis: two-sided
#>