Gradient of the Negative Log-Likelihood for the Kumaraswamy (Kw) Distribution

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the two-parameter Kumaraswamy (Kw) distribution with parameters alpha ($\alpha$) and beta ($\beta$). This provides the analytical gradient often used for efficient optimization via maximum likelihood estimation.

Usage

grkw(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 numeric vector of length 2 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta)$. Returns a vector of NaN if any parameter values are invalid according to their constraints, or if any value in data is not in the interval (0, 1).

Details

The components of the gradient vector of the negative log-likelihood ($-\nabla \ell(\theta | \mathbf{x})$) for the Kw model are:

$$ -\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i) + (\beta-1)\sum_{i=1}^{n}\frac{x_i^{\alpha}\ln(x_i)}{v_i} $$ $$ -\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - \sum_{i=1}^{n}\ln(v_i) $$

where $v_i = 1 - x_i^{\alpha}$. These formulas represent the derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the relevant components of the general GKw gradient (grgkw) 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

(Note: Specific gradient formulas might be derived or sourced from additional references).

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rkw(200, alpha = 2, beta = 3)
par <- c(alpha = 2, beta = 3)

## Gradient of the negative log-likelihood llkw(), not of the log-likelihood
g <- grkw(par, x)
g
#> [1] -8.7375184 -0.9637925

## A small step against the gradient lowers llkw()
llkw(par - 1e-4 * g, x) < llkw(par, x)
#> [1] TRUE

## Agrees with a numerical derivative of llkw()
if (requireNamespace("numDeriv", quietly = TRUE))
  all.equal(g, numDeriv::grad(llkw, par, data = x), tolerance = 1e-6)
#> [1] TRUE