Gradient of the Negative Log-Likelihood for the BKw Distribution

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), gamma ($\gamma$), and delta ($\delta$). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\lambda = 1$. The gradient is typically used in optimization algorithms for maximum likelihood estimation.

Usage

grbkw(par, data)

Arguments

par: A numeric vector of length 4 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), gamma ($\gamma > 0$), delta ($\delta \ge 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a numeric vector of length 4 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \gamma, -\partial \ell/\partial \delta)$. 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 BKw ($\lambda=1$) model are:

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

where:

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

These formulas represent the derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the general GKw gradient (grgkw) components for $\alpha, \beta, \gamma, \delta$ evaluated at $\lambda=1$. Note that the component for $\lambda$ is omitted. Numerical stability is maintained through careful implementation.

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

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

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rbkw(200, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)
par <- c(alpha = 2, beta = 3, gamma = 1.5, delta = 0.5)

## Gradient of the negative log-likelihood llbkw(), not of the log-likelihood
g <- grbkw(par, x)
g
#> [1] -1.6023655  0.4745672 -3.0711866  1.1604209

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

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