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

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

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)

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).

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++.

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).

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

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

Examples

# 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.")
}

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