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

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

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)

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).

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.

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).

Author(s)

Lopes, J. E.

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

# 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")
}

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