Gradient of the Negative Log-Likelihood for the GKw Distribution

Computes the gradient vector (vector of partial derivatives) of the negative log-likelihood function for the five-parameter Generalized Kumaraswamy (GKw) distribution. This provides the analytical gradient, often used for efficient optimization via maximum likelihood estimation.

Usage

grgkw(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 numeric vector of length 5 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \gamma, -\partial \ell/\partial \delta, -\partial \ell/\partial \lambda)$. 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 ($-\nabla \ell(\theta | \mathbf{x})$) are:

$$ -\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i) + \sum_{i=1}^{n}\left[x_i^{\alpha} \ln(x_i) \left(\frac{\beta-1}{v_i} - \frac{(\gamma\lambda-1) \beta v_i^{\beta-1}}{w_i} + \frac{\delta \lambda \beta v_i^{\beta-1} w_i^{\lambda-1}}{z_i}\right)\right] $$ $$ -\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - \sum_{i=1}^{n}\ln(v_i) + \sum_{i=1}^{n}\left[v_i^{\beta} \ln(v_i) \left(\frac{\gamma\lambda-1}{w_i} - \frac{\delta \lambda w_i^{\lambda-1}}{z_i}\right)\right] $$ $$ -\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] - \lambda\sum_{i=1}^{n}\ln(w_i) $$ $$ -\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] - \sum_{i=1}^{n}\ln(z_i) $$ $$ -\frac{\partial \ell}{\partial \lambda} = -\frac{n}{\lambda} - \gamma\sum_{i=1}^{n}\ln(w_i) + \delta\sum_{i=1}^{n}\frac{w_i^{\lambda}\ln(w_i)}{z_i} $$

where:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$
$z_i = 1 - w_i^{\lambda} = 1 - [1-(1-x_i^{\alpha})^{\beta}]^{\lambda}$
$\psi(\cdot)$ is the digamma function (digamma).

Numerical stability is ensured through careful implementation, including checks for valid inputs and handling of intermediate calculations involving potentially small or large numbers, often leveraging the Armadillo C++ library for efficiency.

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(200, 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)

## Gradient of the negative log-likelihood llgkw(), not of the log-likelihood
g <- grgkw(par, x)
g
#> [1] -1.6713047  0.4076473 -3.0711866  1.1604209 -3.8057074

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

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