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

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

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)

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).

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.

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).

Author(s)

Lopes, J. E.

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).

See Also

grgkw (parent distribution gradient), llekw (negative log-likelihood for EKw), hsekw (Hessian for EKw, if available), dekw (density for EKw), optim, grad (for numerical gradient comparison).

Examples

# 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)))

}

gkwreg documentation built on April 16, 2025, 1:10 a.m.