Negative Log-Likelihood for the Kumaraswamy (Kw) Distribution

Computes the negative log-likelihood function for the two-parameter Kumaraswamy (Kw) distribution with parameters alpha ($\alpha$) and beta ($\beta$), given a vector of observations. This function is suitable for maximum likelihood estimation.

Usage

llkw(par, data)

Arguments

par: A numeric vector of length 2 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$).
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 Kumaraswamy (Kw) distribution's probability density function (PDF) is (see dkw): $$ f(x | \theta) = \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} $$ for $0 < x < 1$ and $\theta = (\alpha, \beta)$. The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = n[\ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i)] $$ where $v_i = 1 - x_i^{\alpha}$. 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 $\gamma=1, \delta=0, \lambda=1$.

References

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

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81. doi:10.1016/j.stamet.2008.04.001

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rkw(1000, alpha = 2, beta = 3)
par <- c(alpha = 2, beta = 3)

## llkw() is the negative log-likelihood, -sum(log f(x))
llkw(par, x)
#> [1] -213.2127
-sum(dkw(x, alpha = 2, beta = 3, log = TRUE))
#> [1] -213.2127

## Maximum likelihood: minimize llkw(), with grkw() as its gradient
start <- gkwgetstartvalues(x, family = "kw")
fit <- optim(start, llkw, grkw, data = x, method = "L-BFGS-B", lower = 1e-4)
fit$convergence  # 0: converged
#> [1] 0
fit$par
#>    alpha     beta 
#> 2.008977 3.059364 
fit$value <= llkw(par, x)  # at least as good as the true values
#> [1] TRUE