Negative Log-Likelihood for the Beta-Kumaraswamy (BKw) Distribution

Computes the negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), gamma ($\gamma$), and delta ($\delta$), given a vector of observations. This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\lambda = 1$. This function is typically used for maximum likelihood estimation via numerical optimization.

Usage

llbkw(par, data)

Arguments

par: A numeric vector of length 4 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), 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

The Beta-Kumaraswamy (BKw) distribution is the GKw distribution (dgkw) with $\lambda=1$. Its probability density function (PDF) is: $$ f(x | \theta) = \frac{\alpha \beta}{B(\gamma, \delta+1)} x^{\alpha - 1} \bigl(1 - x^\alpha\bigr)^{\beta(\delta+1) - 1} \bigl[1 - \bigl(1 - x^\alpha\bigr)^\beta\bigr]^{\gamma - 1} $$ for $0 < x < 1$, $\theta = (\alpha, \beta, \gamma, \delta)$, 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 $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = n[\ln(\alpha) + \ln(\beta) - \ln B(\gamma, \delta+1)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta(\delta+1)-1)\ln(v_i) + (\gamma-1)\ln(w_i)] $$ where:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$

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 similarly to llgkw.

References

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 <- rbkw(1000, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)
par <- c(alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)

## llbkw() is the negative log-likelihood, -sum(log f(x))
llbkw(par, x)
#> [1] -351.0014
-sum(dbkw(x, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, log = TRUE))
#> [1] -351.0014

## Maximum likelihood: minimize llbkw(), with grbkw() as its gradient
start <- gkwgetstartvalues(x, family = "bkw")
fit <- optim(start, llbkw, grbkw, data = x, method = "L-BFGS-B", lower = 1e-4)
fit$convergence  # 0: converged
#> [1] 0
## The parameters of this family are weakly identified: compare likelihoods
fit$value <= llbkw(par, x)  # at least as good as the true values
#> [1] TRUE