grekw: Gradient of the Negative Log-Likelihood for the EKw Distribution

Description

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), and lambda ($\lambda$). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\gamma = 1$ and $\delta = 0$. The gradient is useful for optimization.

Usage

grekw(par, data)

Value

Returns a numeric vector of length 3 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \lambda)$. 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).

Arguments

par: A numeric vector of length 3 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), lambda ($\lambda > 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Author

Lopes, J. E.

Details

The components of the gradient vector of the negative log-likelihood ($-\nabla \ell(\theta | \mathbf{x})$) for the EKw ($\gamma=1, \delta=0$) model are:

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

where:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$

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 $\gamma=1, \delta=0$.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation,

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

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

Examples

Run this code

# \donttest{
# Assuming existence of rekw, llekw, grekw, hsekw functions for EKw

# Generate sample data
set.seed(123)
true_par_ekw <- c(alpha = 2, beta = 3, lambda = 0.5)
if (exists("rekw")) {
  sample_data_ekw <- rekw(100, alpha = true_par_ekw[1], beta = true_par_ekw[2],
                          lambda = true_par_ekw[3])
} else {
  sample_data_ekw <- rgkw(100, alpha = true_par_ekw[1], beta = true_par_ekw[2],
                          gamma = 1, delta = 0, lambda = true_par_ekw[3])
}
hist(sample_data_ekw, breaks = 20, main = "EKw(2, 3, 0.5) Sample")

# --- Find MLE estimates ---
start_par_ekw <- c(1.5, 2.5, 0.8)
mle_result_ekw <- stats::optim(par = start_par_ekw,
                               fn = llekw,
                               gr = grekw, # Use analytical gradient for EKw
                               method = "BFGS",
                               hessian = TRUE,
                               data = sample_data_ekw)

# --- Compare analytical gradient to numerical gradient ---
if (mle_result_ekw$convergence == 0 &&
    requireNamespace("numDeriv", quietly = TRUE)) {

  mle_par_ekw <- mle_result_ekw$par
  cat("\nComparing Gradients for EKw at MLE estimates:\n")

  # Numerical gradient of llekw
  num_grad_ekw <- numDeriv::grad(func = llekw, x = mle_par_ekw, data = sample_data_ekw)

  # Analytical gradient from grekw
  ana_grad_ekw <- grekw(par = mle_par_ekw, data = sample_data_ekw)

  cat("Numerical Gradient (EKw):\n")
  print(num_grad_ekw)
  cat("Analytical Gradient (EKw):\n")
  print(ana_grad_ekw)

  # Check differences
  cat("Max absolute difference between EKw gradients:\n")
  print(max(abs(num_grad_ekw - ana_grad_ekw)))

} else {
  cat("\nSkipping EKw gradient comparison.\n")
}

# Example with Hessian comparison (if hsekw exists)
if (mle_result_ekw$convergence == 0 &&
    requireNamespace("numDeriv", quietly = TRUE) && exists("hsekw")) {

  num_hess_ekw <- numDeriv::hessian(func = llekw, x = mle_par_ekw, data = sample_data_ekw)
  ana_hess_ekw <- hsekw(par = mle_par_ekw, data = sample_data_ekw)
  cat("\nMax absolute difference between EKw Hessians:\n")
  print(max(abs(num_hess_ekw - ana_hess_ekw)))

}

# }

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