Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization)

Computes the negative log-likelihood function for the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by gamma ($\gamma$) and delta ($\delta$), corresponding to the standard Beta distribution with shape parameters shape1 = gamma and shape2 = delta + 1. This function is suitable for maximum likelihood estimation.

Usage

llbeta(par, data)

Arguments

par: A numeric vector of length 2 containing the distribution parameters in the order: gamma ($\gamma > 0$), delta ($\delta \ge 0$).
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

This function calculates the negative log-likelihood for a Beta distribution with parameters shape1 = gamma ($\gamma$) and shape2 = delta + 1 ($\delta+1$). The probability density function (PDF) is: $$ f(x | \gamma, \delta) = \frac{x^{\gamma-1} (1-x)^{\delta}}{B(\gamma, \delta+1)} $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (beta). The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = \sum_{i=1}^{n} [(\gamma-1)\ln(x_i) + \delta\ln(1-x_i)] - n \ln B(\gamma, \delta+1) $$ where $\theta = (\gamma, \delta)$. This function computes and returns the negative log-likelihood, $-\ell(\theta|\mathbf{x})$, suitable for minimization using optimization routines like optim. It is equivalent to the negative log-likelihood of the GKw distribution (llgkw) evaluated at $\alpha=1, \beta=1, \lambda=1$, and also to the negative log-likelihood of the McDonald distribution (llmc) evaluated at $\lambda=1$. The term $\ln B(\gamma, \delta+1)$ is typically computed using log-gamma functions (lgamma) for numerical stability.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd ed.). Wiley.

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

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rbeta_(1000, gamma = 2, delta = 3)
par <- c(gamma = 2, delta = 3)

## llbeta() is the negative log-likelihood, -sum(log f(x))
llbeta(par, x)
#> [1] -359.6415
-sum(dbeta_(x, gamma = 2, delta = 3, log = TRUE))
#> [1] -359.6415

## Maximum likelihood: minimize llbeta(), with grbeta() as its gradient
start <- gkwgetstartvalues(x, family = "beta")
fit <- optim(start, llbeta, grbeta, data = x, method = "L-BFGS-B", lower = 1e-4)
fit$convergence  # 0: converged
#> [1] 0
fit$par
#>    gamma    delta 
#> 2.028741 2.997411 
fit$value <= llbeta(par, x)  # at least as good as the true values
#> [1] TRUE