Negative Log-Likelihood for the Generalized Kumaraswamy Distribution

Computes the negative log-likelihood function for the five-parameter Generalized Kumaraswamy (GKw) distribution given a vector of observations. This function is designed for use in optimization routines (e.g., maximum likelihood estimation).

Usage

llgkw(par, data)

Arguments

par: A numeric vector of length 5 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), 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 probability density function (PDF) of the GKw distribution is given in dgkw. The log-likelihood function $\ell(\theta)$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is: $$ \ell(\theta | \mathbf{x}) = n\ln(\lambda\alpha\beta) - n\ln B(\gamma,\delta+1) + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\gamma\lambda-1)\ln(w_i) + \delta\ln(z_i)] $$ where $\theta = (\alpha, \beta, \gamma, \delta, \lambda)$, $B(a,b)$ is the Beta function (beta), and:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$
$z_i = 1 - w_i^{\lambda} = 1 - [1-(1-x_i^{\alpha})^{\beta}]^{\lambda}$

This function computes $-\ell(\theta|\mathbf{x})$.

Numerical stability is prioritized using:

lbeta function for the log-Beta term.
Log-transformations of intermediate terms ($v_i, w_i, z_i$) and use of log1p where appropriate to handle values close to 0 or 1 accurately.
Checks for invalid parameters and data.

References

Carrasco, J. M. F., Ferrari, S. L. P., & Cordeiro, G. M. (2010). A new generalized Kumaraswamy distribution. arXiv preprint arXiv:1004.0911. doi:10.48550/arXiv.1004.0911

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 <- rgkw(1000, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)
par <- c(alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)

## llgkw() is the negative log-likelihood, -sum(log f(x))
llgkw(par, x)
#> [1] -392.775
-sum(dgkw(x, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2,
    log = TRUE))
#> [1] -392.775

## Maximum likelihood: minimize llgkw(), with grgkw() as its gradient
start <- gkwgetstartvalues(x, family = "gkw")
fit <- optim(start, llgkw, grgkw, 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 <= llgkw(par, x)  # at least as good as the true values
#> [1] TRUE