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

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

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)

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).

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.

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).

Author(s)

Lopes, J. E.

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

# 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")
}

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