gkwreg (version 1.0.7)

hsgkw: Hessian Matrix of the Negative Log-Likelihood for the GKw Distribution

Description

Computes the analytic Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the five-parameter Generalized Kumaraswamy (GKw) distribution. This is typically used to estimate standard errors of maximum likelihood estimates or in optimization algorithms.

Usage

hsgkw(par, data)

Value

Returns a 5x5 numeric matrix representing the Hessian matrix of the negative log-likelihood function, i.e., the matrix of second partial derivatives \(-\partial^2 \ell / (\partial \theta_i \partial \theta_j)\). Returns a 5x5 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 5 containing the distribution parameters in the order: alpha (\(\alpha > 0\)), beta (\(\beta > 0\)), gamma (\(\gamma > 0\)), 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 GKw PDF (see dgkw). The log-likelihood function \(\ell(\theta | \mathbf{x})\) is given by: $$ \ell(\theta) = n \ln(\lambda\alpha\beta) - n \ln B(\gamma, \delta+1) + \sum_{i=1}^{n} [(\alpha-1) \ln(x_i) + (\beta-1) \ln(v_i) + (\gamma\lambda - 1) \ln(w_i) + \delta \ln(z_i)] $$ where \(\theta = (\alpha, \beta, \gamma, \delta, \lambda)\), \(B(a,b)\) is the Beta function (beta), 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 \(- \frac{\partial^2 \ell(\theta | \mathbf{x})}{\partial \theta_i \partial \theta_j}\).

Key properties of the returned matrix:

• Dimensions: 5x5.

• Symmetry: The matrix is symmetric.

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

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

The exact analytical formulas for the second derivatives are implemented directly (often derived using symbolic differentiation) for accuracy and efficiency, typically using C++.

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

llgkw (negative log-likelihood function), grgkw (gradient vector), dgkw (density function), gkwreg (if provides regression fitting), optim, hessian (for numerical Hessian comparison).

Examples

# \donttest{
# Generate sample data from a known GKw distribution
set.seed(123)
true_par <- c(alpha = 2, beta = 3, gamma = 1.0, delta = 0.5, lambda = 0.5)
sample_data <- rgkw(100, alpha = true_par[1], beta = true_par[2],
                   gamma = true_par[3], delta = true_par[4], lambda = true_par[5])

# --- Find MLE estimates (e.g., using optim) ---
start_par <- c(1.5, 2.5, 1.2, 0.3, 0.6) # Initial guess
mle_result <- stats::optim(par = start_par,
                           fn = llgkw,
                           method = "BFGS",
                           hessian = TRUE, # Ask optim for numerical Hessian
                           data = sample_data)

if (mle_result$convergence == 0) {
  mle_par <- mle_result$par
  print("MLE parameters found:")
  print(mle_par)

  # --- Compare analytical Hessian to numerical Hessian ---
  # Requires the 'numDeriv' package
  if (requireNamespace("numDeriv", quietly = TRUE)) {

    cat("\nComparing Hessians at MLE estimates:\n")

    # Numerical Hessian from numDeriv applied to negative LL function
    num_hess <- numDeriv::hessian(func = llgkw, x = mle_par, data = sample_data)

    # Analytical Hessian (output of hsgkw)
    ana_hess <- hsgkw(par = mle_par, data = sample_data)

    cat("Numerical Hessian (from numDeriv):\n")
    print(round(num_hess, 4))
    cat("Analytical Hessian (from hsgkw):\n")
    print(round(ana_hess, 4))

    # Check differences (should be small)
    cat("Max absolute difference between Hessians:\n")
    print(max(abs(num_hess - ana_hess)))

    # Optional: Use analytical Hessian to estimate standard errors
    # Ensure Hessian is positive definite before inverting
    # fisher_info_analytic <- ana_hess # Hessian of negative LL
    # tryCatch({
    #   cov_matrix <- solve(fisher_info_analytic)
    #   std_errors <- sqrt(diag(cov_matrix))
    #   cat("Std. Errors from Analytical Hessian:\n")
    #   print(std_errors)
    # }, error = function(e) {
    #   warning("Could not invert analytical Hessian to get standard errors: ", e$message)
    # })

  } else {
    cat("\nSkipping Hessian comparison (requires 'numDeriv' package).\n")
  }
} else {
  warning("Optimization did not converge. Hessian comparison skipped.")
}

# }