Negative Log-Likelihood for the KKw Distribution

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

Usage

llkkw(par, data)

Arguments

par: A numeric vector of length 4 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), 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 KKw distribution is the GKw distribution (dgkw) with $\gamma=1$. Its probability density function (PDF) is: $$ f(x | \theta) = (\delta + 1) \lambda \alpha \beta x^{\alpha - 1} (1 - x^\alpha)^{\beta - 1} \bigl[1 - (1 - x^\alpha)^\beta\bigr]^{\lambda - 1} \bigl\{1 - \bigl[1 - (1 - x^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta} $$ for $0 < x < 1$ and $\theta = (\alpha, \beta, \delta, \lambda)$. The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = n[\ln(\delta+1) + \ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i) + \delta\ln(z_i)] $$ where:

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

This function computes 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 <- rkkw(1000, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2)
par <- c(alpha = 2, beta = 3, delta = 0.5, lambda = 1.2)

## llkkw() is the negative log-likelihood, -sum(log f(x))
llkkw(par, x)
#> [1] -349.5218
-sum(dkkw(x, alpha = 2, beta = 3, delta = 0.5, lambda = 1.2, log = TRUE))
#> [1] -349.5218

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