gkwreg (version 1.0.7)

hskw: Hessian Matrix of the Negative Log-Likelihood for the Kw Distribution

Description

Computes the analytic 2x2 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the two-parameter Kumaraswamy (Kw) distribution with parameters alpha (\(\alpha\)) and beta (\(\beta\)). The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hskw(par, data)

Value

Returns a 2x2 numeric matrix representing the Hessian matrix of the negative log-likelihood function, \(-\partial^2 \ell / (\partial \theta_i \partial \theta_j)\), where \(\theta = (\alpha, \beta)\). Returns a 2x2 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 2 containing the distribution parameters in the order: alpha (\(\alpha > 0\)), beta (\(\beta > 0\)).

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 (\(-\ell(\theta|\mathbf{x})\)). The components are the negative of the second derivatives of the log-likelihood \(\ell\) (derived from the PDF in dkw).

Let \(v_i = 1 - x_i^{\alpha}\). The second derivatives of the positive log-likelihood (\(\ell\)) are: $$ \frac{\partial^2 \ell}{\partial \alpha^2} = -\frac{n}{\alpha^2} - (\beta-1)\sum_{i=1}^{n}\frac{x_i^{\alpha}(\ln(x_i))^2}{v_i^2} $$ $$ \frac{\partial^2 \ell}{\partial \alpha \partial \beta} = - \sum_{i=1}^{n}\frac{x_i^{\alpha}\ln(x_i)}{v_i} $$ $$ \frac{\partial^2 \ell}{\partial \beta^2} = -\frac{n}{\beta^2} $$ The function returns the Hessian matrix containing the negative of these values.

Key properties of the returned matrix:

• Dimensions: 2x2.

• Symmetry: The matrix is symmetric.

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

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

This corresponds to the relevant 2x2 submatrix of the 5x5 GKw Hessian (hsgkw) evaluated at \(\gamma=1, \delta=0, \lambda=1\).

References

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

(Note: Specific Hessian formulas might be derived or sourced from additional references).

See Also

hsgkw (parent distribution Hessian), llkw (negative log-likelihood for Kw), grkw (gradient for Kw, if available), dkw (density for Kw), optim, hessian (for numerical Hessian comparison).

Examples

# \donttest{
# Assuming existence of rkw, llkw, grkw, hskw functions for Kw

# Generate sample data
set.seed(123)
true_par_kw <- c(alpha = 2, beta = 3)
sample_data_kw <- rkw(100, alpha = true_par_kw[1], beta = true_par_kw[2])
hist(sample_data_kw, breaks = 20, main = "Kw(2, 3) Sample")

# --- Find MLE estimates ---
start_par_kw <- c(1.5, 2.5)
mle_result_kw <- stats::optim(par = start_par_kw,
                              fn = llkw,
                              gr = if (exists("grkw")) grkw else NULL,
                              method = "L-BFGS-B",
                              lower = c(1e-6, 1e-6),
                              hessian = TRUE, # Ask optim for numerical Hessian
                              data = sample_data_kw)

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

  mle_par_kw <- mle_result_kw$par
  cat("\nComparing Hessians for Kw at MLE estimates:\n")

  # Numerical Hessian of llkw
  num_hess_kw <- numDeriv::hessian(func = llkw, x = mle_par_kw, data = sample_data_kw)

  # Analytical Hessian from hskw
  ana_hess_kw <- hskw(par = mle_par_kw, data = sample_data_kw)

  cat("Numerical Hessian (Kw):\n")
  print(round(num_hess_kw, 4))
  cat("Analytical Hessian (Kw):\n")
  print(round(ana_hess_kw, 4))

  # Check differences
  cat("Max absolute difference between Kw Hessians:\n")
  print(max(abs(num_hess_kw - ana_hess_kw)))

  # Optional: Use analytical Hessian for Standard Errors
  # tryCatch({
  #   cov_matrix_kw <- solve(ana_hess_kw) # ana_hess_kw is already Hessian of negative LL
  #   std_errors_kw <- sqrt(diag(cov_matrix_kw))
  #   cat("Std. Errors from Analytical Kw Hessian:\n")
  #   print(std_errors_kw)
  # }, error = function(e) {
  #   warning("Could not invert analytical Kw Hessian: ", e$message)
  # })

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

# }