gkwreg (version 1.0.7)

hsekw: Hessian Matrix of the Negative Log-Likelihood for the EKw Distribution

Description

Computes the analytic 3x3 Hessian matrix (matrix of second 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 Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hsekw(par, data)

Value

Returns a 3x3 numeric matrix representing the Hessian matrix of the negative log-likelihood function, \(-\partial^2 \ell / (\partial \theta_i \partial \theta_j)\), where \(\theta = (\alpha, \beta, \lambda)\). Returns a 3x3 matrix populated with 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

This function calculates the analytic second partial derivatives of the negative log-likelihood function based on the EKw log-likelihood (\(\gamma=1, \delta=0\) case of GKw, see llekw): $$ \ell(\theta | \mathbf{x}) = n[\ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i)] $$ where \(\theta = (\alpha, \beta, \lambda)\) and intermediate terms are:

• \(v_i = 1 - x_i^{\alpha}\)

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

The Hessian matrix returned contains the elements \(- \frac{\partial^2 \ell(\theta | \mathbf{x})}{\partial \theta_i \partial \theta_j}\) for \(\theta_i, \theta_j \in \{\alpha, \beta, \lambda\}\).

Key properties of the returned matrix:

• Dimensions: 3x3.

• Symmetry: The matrix is symmetric.

• Ordering: Rows and columns correspond to the parameters in the order \(\alpha, \beta, \lambda\).

• Content: Analytic second derivatives of the negative log-likelihood.

This corresponds to the relevant 3x3 submatrix of the 5x5 GKw Hessian (hsgkw) evaluated at \(\gamma=1, \delta=0\). The exact analytical formulas are implemented directly.

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 Hessian formulas might be derived or sourced from additional references).

See Also

hsgkw (parent distribution Hessian), llekw (negative log-likelihood for EKw), grekw (gradient for EKw, if available), dekw (density for EKw), optim, hessian (for numerical Hessian comparison).

Examples

# \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 = if (exists("grekw")) grekw else NULL,
                               method = "BFGS",
                               hessian = TRUE, # Ask optim for numerical Hessian
                               data = sample_data_ekw)

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

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

  # Numerical Hessian of llekw
  num_hess_ekw <- numDeriv::hessian(func = llekw, x = mle_par_ekw, data = sample_data_ekw)

  # Analytical Hessian from hsekw
  ana_hess_ekw <- hsekw(par = mle_par_ekw, data = sample_data_ekw)

  cat("Numerical Hessian (EKw):\n")
  print(round(num_hess_ekw, 4))
  cat("Analytical Hessian (EKw):\n")
  print(round(ana_hess_ekw, 4))

  # Check differences
  cat("Max absolute difference between EKw Hessians:\n")
  print(max(abs(num_hess_ekw - ana_hess_ekw)))

  # Optional: Use analytical Hessian for Standard Errors
  # tryCatch({
  #   cov_matrix_ekw <- solve(ana_hess_ekw)
  #   std_errors_ekw <- sqrt(diag(cov_matrix_ekw))
  #   cat("Std. Errors from Analytical EKw Hessian:\n")
  #   print(std_errors_ekw)
  # }, error = function(e) {
  #   warning("Could not invert analytical EKw Hessian: ", e$message)
  # })

} else {
  cat("\nSkipping EKw Hessian comparison.\n")
  cat("Requires convergence, 'numDeriv' package, and function 'hsekw'.\n")
}

# }