gkwreg (version 1.0.7)

hskkw: Hessian Matrix of the Negative Log-Likelihood for the kkw Distribution

Description

Computes the analytic 4x4 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the Kumaraswamy-Kumaraswamy (kkw) distribution with parameters alpha (\(\alpha\)), beta (\(\beta\)), delta (\(\delta\)), and lambda (\(\lambda\)). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where \(\gamma = 1\). The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hskkw(par, data)

Value

Returns a 4x4 numeric matrix representing the Hessian matrix of the negative log-likelihood function, \(-\partial^2 \ell / (\partial \theta_i \partial \theta_j)\), where \(\theta = (\alpha, \beta, \delta, \lambda)\). Returns a 4x4 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 4 containing the distribution parameters in the order: alpha (\(\alpha > 0\)), beta (\(\beta > 0\)), delta (\(\delta \ge 0\)), 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 kkw log-likelihood (\(\gamma=1\) case of GKw, see llkkw): $$ \ell(\theta | \mathbf{x}) = n[\ln(\delta+1) + \ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i) + \delta\ln(z_i)] $$ where \(\theta = (\alpha, \beta, \delta, \lambda)\) and intermediate terms are:

• \(v_i = 1 - x_i^{\alpha}\)

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

• \(z_i = 1 - w_i^{\lambda} = 1 - [1-(1-x_i^{\alpha})^{\beta}]^{\lambda}\)

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

Key properties of the returned matrix:

• Dimensions: 4x4.

• Symmetry: The matrix is symmetric.

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

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

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

References

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.

See Also

hsgkw (parent distribution Hessian), llkkw (negative log-likelihood for kkw), grkkw (gradient for kkw), dkkw (density for kkw), optim, hessian (for numerical Hessian comparison).

Examples

# \donttest{
# Assuming existence of rkkw, llkkw, grkkw, hskkw functions for kkw

# Generate sample data
set.seed(123)
true_par_kkw <- c(alpha = 2, beta = 3, delta = 1.5, lambda = 0.5)
if (exists("rkkw")) {
  sample_data_kkw <- rkkw(100, alpha = true_par_kkw[1], beta = true_par_kkw[2],
                         delta = true_par_kkw[3], lambda = true_par_kkw[4])
} else {
  sample_data_kkw <- rgkw(100, alpha = true_par_kkw[1], beta = true_par_kkw[2],
                         gamma = 1, delta = true_par_kkw[3], lambda = true_par_kkw[4])
}

# --- Find MLE estimates ---
start_par_kkw <- c(1.5, 2.5, 1.0, 0.6)
mle_result_kkw <- stats::optim(par = start_par_kkw,
                               fn = llkkw,
                               gr = if (exists("grkkw")) grkkw else NULL,
                               method = "BFGS",
                               hessian = TRUE,
                               data = sample_data_kkw)

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

  mle_par_kkw <- mle_result_kkw$par
  cat("\nComparing Hessians for kkw at MLE estimates:\n")

  # Numerical Hessian of llkkw
  num_hess_kkw <- numDeriv::hessian(func = llkkw, x = mle_par_kkw, data = sample_data_kkw)

  # Analytical Hessian from hskkw
  ana_hess_kkw <- hskkw(par = mle_par_kkw, data = sample_data_kkw)

  cat("Numerical Hessian (kkw):\n")
  print(round(num_hess_kkw, 4))
  cat("Analytical Hessian (kkw):\n")
  print(round(ana_hess_kkw, 4))

  # Check differences
  cat("Max absolute difference between kkw Hessians:\n")
  print(max(abs(num_hess_kkw - ana_hess_kkw)))

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

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

# }