Negative Log-Likelihood for the Exponentiated Kumaraswamy (EKw) Distribution

Computes the negative log-likelihood function for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), and lambda ($\lambda$), given a vector of observations. This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\gamma = 1$ and $\delta = 0$. This function is suitable for maximum likelihood estimation.

Usage

llekw(par, data)

Arguments

par: A numeric vector of length 3 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), 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 Exponentiated Kumaraswamy (EKw) distribution is the GKw distribution (dekw) with $\gamma=1$ and $\delta=0$. Its probability density function (PDF) is: $$ f(x | \theta) = \lambda \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} \bigl[1 - (1 - x^\alpha)^\beta \bigr]^{\lambda - 1} $$ for $0 < x < 1$ and $\theta = (\alpha, \beta, \lambda)$. The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = n[\ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i)] $$ 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

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

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 <- rekw(1000, alpha = 2, beta = 3, lambda = 1.2)
par <- c(alpha = 2, beta = 3, lambda = 1.2)

## llekw() is the negative log-likelihood, -sum(log f(x))
llekw(par, x)
#> [1] -245.6122
-sum(dekw(x, alpha = 2, beta = 3, lambda = 1.2, log = TRUE))
#> [1] -245.6122

## Maximum likelihood: minimize llekw(), with grekw() as its gradient
start <- gkwgetstartvalues(x, family = "ekw")
fit <- optim(start, llekw, grekw, data = x, method = "L-BFGS-B", lower = 1e-4)
fit$convergence  # 0: converged
#> [1] 0
fit$par
#>    alpha     beta   lambda 
#> 2.110857 3.121146 1.125558 
fit$value <= llekw(par, x)  # at least as good as the true values
#> [1] TRUE