Negative Log-Likelihood for the McDonald (Mc)/Beta Power Distribution

Computes the negative log-likelihood function for the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma ($\gamma$), delta ($\delta$), and lambda ($\lambda$), given a vector of observations. This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\alpha = 1$ and $\beta = 1$. This function is suitable for maximum likelihood estimation.

Usage

llmc(par, data)

Arguments

par: A numeric vector of length 3 containing the distribution parameters in the order: gamma ($\gamma > 0$), delta ($\delta \ge 0$), lambda ($\lambda > 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a single double value representing the negative log-likelihood ($-\ell(\theta|\mathbf{x})$). Returns Inf if any parameter values in par are invalid according to their constraints, or if any value in data is not in the interval (0, 1); in the latter case a warning naming data is also signaled, because an infinite objective offers an optimizer no gradient direction to follow and more often means a sample on the wrong scale than a genuine fit failure.

Details

The McDonald (Mc) distribution is the GKw distribution (dmc) with $\alpha=1$ and $\beta=1$. Its probability density function (PDF) is: $$ f(x | \theta) = \frac{\lambda}{B(\gamma,\delta+1)} x^{\gamma \lambda - 1} (1 - x^\lambda)^\delta $$ for $0 < x < 1$, $\theta = (\gamma, \delta, \lambda)$, and $B(a,b)$ is the Beta function (beta). The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = n[\ln(\lambda) - \ln B(\gamma, \delta+1)] + \sum_{i=1}^{n} [(\gamma\lambda - 1)\ln(x_i) + \delta\ln(1 - x_i^\lambda)] $$ This function computes and returns the negative log-likelihood, $-\ell(\theta|\mathbf{x})$, suitable for minimization using optimization routines like optim. Numerical stability is maintained, including using the log-gamma function (lgamma) for the Beta function term.

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

set.seed(123)
x <- rmc(1000, gamma = 0.5, delta = 5, lambda = 3)
par <- c(gamma = 0.5, delta = 5, lambda = 3)

## llmc() is the negative log-likelihood, -sum(log f(x))
llmc(par, x)
#> [1] -349.7525
-sum(dmc(x, gamma = 0.5, delta = 5, lambda = 3, log = TRUE))
#> [1] -349.7525

## Maximum likelihood: minimize llmc(), with grmc() as its gradient
start <- gkwgetstartvalues(x, family = "mc")
fit <- optim(start, llmc, grmc, data = x, method = "L-BFGS-B", lower = 1e-4)
fit$convergence  # 0: converged
#> [1] 0
fit$par
#>     gamma     delta    lambda 
#> 0.4712336 5.2348706 3.0191417 
fit$value <= llmc(par, x)  # at least as good as the true values
#> [1] TRUE