hsbeta: Hessian Matrix of the Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization)

Description

Computes the analytic 2x2 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by gamma ($\gamma$) and delta ($\delta$), corresponding to the standard Beta distribution with shape parameters shape1 = gamma and shape2 = delta + 1. The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hsbeta(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 $\theta = (\gamma, \delta)$. 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: gamma ($\gamma > 0$), delta ($\delta \ge 0$).
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 ($-\ell(\theta|\mathbf{x})$) for a Beta distribution with parameters shape1 = gamma ($\gamma$) and shape2 = delta + 1 ($\delta+1$). The components of the Hessian matrix ($-\mathbf{H}(\theta)$) are:

$$ -\frac{\partial^2 \ell}{\partial \gamma^2} = n[\psi'(\gamma) - \psi'(\gamma+\delta+1)] $$ $$ -\frac{\partial^2 \ell}{\partial \gamma \partial \delta} = -n\psi'(\gamma+\delta+1) $$ $$ -\frac{\partial^2 \ell}{\partial \delta^2} = n[\psi'(\delta+1) - \psi'(\gamma+\delta+1)] $$

where $\psi'(\cdot)$ is the trigamma function (trigamma). These formulas represent the second derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the relevant 2x2 submatrix of the general GKw Hessian (hsgkw) evaluated at $\alpha=1, \beta=1, \lambda=1$. Note the parameterization difference from the standard Beta distribution (shape2 = delta + 1).

The returned matrix is symmetric.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd ed.). Wiley.

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation,

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

Examples

Run this code

# \donttest{
## Example 1: Basic Hessian Evaluation

# Generate sample data
set.seed(123)
n <- 1000
true_params <- c(gamma = 2.0, delta = 3.0)
data <- rbeta_(n, gamma = true_params[1], delta = true_params[2])

# Evaluate Hessian at true parameters
hess_true <- hsbeta(par = true_params, data = data)
cat("Hessian matrix at true parameters:\n")
print(hess_true, digits = 4)

# Check symmetry
cat("\nSymmetry check (max |H - H^T|):",
    max(abs(hess_true - t(hess_true))), "\n")


## Example 2: Hessian Properties at MLE

# Fit model
fit <- optim(
  par = c(1.5, 2.5),
  fn = llbeta,
  gr = grbeta,
  data = data,
  method = "L-BFGS-B",
  lower = c(0.01, 0.01),
  upper = c(100, 100),
  hessian = TRUE
)

mle <- fit$par
names(mle) <- c("gamma", "delta")

# Hessian at MLE
hessian_at_mle <- hsbeta(par = mle, data = data)
cat("\nHessian at MLE:\n")
print(hessian_at_mle, digits = 4)

# Compare with optim's numerical Hessian
cat("\nComparison with optim Hessian:\n")
cat("Max absolute difference:",
    max(abs(hessian_at_mle - fit$hessian)), "\n")

# Eigenvalue analysis
eigenvals <- eigen(hessian_at_mle, only.values = TRUE)$values
cat("\nEigenvalues:\n")
print(eigenvals)

cat("\nPositive definite:", all(eigenvals > 0), "\n")
cat("Condition number:", max(eigenvals) / min(eigenvals), "\n")


## Example 3: Standard Errors and Confidence Intervals

# Observed information matrix
obs_info <- hessian_at_mle

# Variance-covariance matrix
vcov_matrix <- solve(obs_info)
cat("\nVariance-Covariance Matrix:\n")
print(vcov_matrix, digits = 6)

# Standard errors
se <- sqrt(diag(vcov_matrix))
names(se) <- c("gamma", "delta")

# Correlation matrix
corr_matrix <- cov2cor(vcov_matrix)
cat("\nCorrelation Matrix:\n")
print(corr_matrix, digits = 4)

# Confidence intervals
z_crit <- qnorm(0.975)
results <- data.frame(
  Parameter = c("gamma", "delta"),
  True = true_params,
  MLE = mle,
  SE = se,
  CI_Lower = mle - z_crit * se,
  CI_Upper = mle + z_crit * se
)
print(results, digits = 4)

cat(sprintf("\nMLE corresponds approx to Beta(%.2f, %.2f)\n",
    mle[1], mle[2] + 1))
cat("True corresponds to Beta(%.2f, %.2f)\n",
    true_params[1], true_params[2] + 1)


## Example 4: Determinant and Trace Analysis

# Compute at different points
test_params <- rbind(
  c(1.5, 2.5),
  c(2.0, 3.0),
  mle,
  c(2.5, 3.5)
)

hess_properties <- data.frame(
  Gamma = numeric(),
  Delta = numeric(),
  Determinant = numeric(),
  Trace = numeric(),
  Min_Eigenval = numeric(),
  Max_Eigenval = numeric(),
  Cond_Number = numeric(),
  stringsAsFactors = FALSE
)

for (i in 1:nrow(test_params)) {
  H <- hsbeta(par = test_params[i, ], data = data)
  eigs <- eigen(H, only.values = TRUE)$values

  hess_properties <- rbind(hess_properties, data.frame(
    Gamma = test_params[i, 1],
    Delta = test_params[i, 2],
    Determinant = det(H),
    Trace = sum(diag(H)),
    Min_Eigenval = min(eigs),
    Max_Eigenval = max(eigs),
    Cond_Number = max(eigs) / min(eigs)
  ))
}

cat("\nHessian Properties at Different Points:\n")
print(hess_properties, digits = 4, row.names = FALSE)


## Example 5: Curvature Visualization (Gamma vs Delta)

# Create grid around MLE
gamma_grid <- seq(mle[1] - 1.5, mle[1] + 1.5, length.out = 25)
delta_grid <- seq(mle[2] - 1.5, mle[2] + 1.5, length.out = 25)
gamma_grid <- gamma_grid[gamma_grid > 0]
delta_grid <- delta_grid[delta_grid > 0]

# Compute curvature measures
determinant_surface <- matrix(NA, nrow = length(gamma_grid),
                               ncol = length(delta_grid))
trace_surface <- matrix(NA, nrow = length(gamma_grid),
                         ncol = length(delta_grid))

for (i in seq_along(gamma_grid)) {
  for (j in seq_along(delta_grid)) {
    H <- hsbeta(c(gamma_grid[i], delta_grid[j]), data)
    determinant_surface[i, j] <- det(H)
    trace_surface[i, j] <- sum(diag(H))
  }
}

# Plot

contour(gamma_grid, delta_grid, determinant_surface,
        xlab = expression(gamma), ylab = expression(delta),
        main = "Hessian Determinant", las = 1,
        col = "#2E4057", lwd = 1.5, nlevels = 15)
points(mle[1], mle[2], pch = 19, col = "#8B0000", cex = 1.5)
points(true_params[1], true_params[2], pch = 17, col = "#006400", cex = 1.5)
grid(col = "gray90")

contour(gamma_grid, delta_grid, trace_surface,
        xlab = expression(gamma), ylab = expression(delta),
        main = "Hessian Trace", las = 1,
        col = "#2E4057", lwd = 1.5, nlevels = 15)
points(mle[1], mle[2], pch = 19, col = "#8B0000", cex = 1.5)
points(true_params[1], true_params[2], pch = 17, col = "#006400", cex = 1.5)
grid(col = "gray90")


## Example 6: Confidence Ellipse (Gamma vs Delta)

# Extract 2x2 submatrix (full matrix in this case)
vcov_2d <- vcov_matrix

# Create confidence ellipse
theta <- seq(0, 2 * pi, length.out = 100)
chi2_val <- qchisq(0.95, df = 2)

eig_decomp <- eigen(vcov_2d)
ellipse <- matrix(NA, nrow = 100, ncol = 2)
for (i in 1:100) {
  v <- c(cos(theta[i]), sin(theta[i]))
  ellipse[i, ] <- mle + sqrt(chi2_val) *
    (eig_decomp$vectors %*% diag(sqrt(eig_decomp$values)) %*% v)
}

# Marginal confidence intervals
se_2d <- sqrt(diag(vcov_2d))
ci_gamma <- mle[1] + c(-1, 1) * 1.96 * se_2d[1]
ci_delta <- mle[2] + c(-1, 1) * 1.96 * se_2d[2]

# Plot

plot(ellipse[, 1], ellipse[, 2], type = "l", lwd = 2, col = "#2E4057",
     xlab = expression(gamma), ylab = expression(delta),
     main = "95% Confidence Ellipse (Gamma vs Delta)", las = 1)

# Add marginal CIs
abline(v = ci_gamma, col = "#808080", lty = 3, lwd = 1.5)
abline(h = ci_delta, col = "#808080", lty = 3, lwd = 1.5)

points(mle[1], mle[2], pch = 19, col = "#8B0000", cex = 1.5)
points(true_params[1], true_params[2], pch = 17, col = "#006400", cex = 1.5)

legend("topright",
       legend = c("MLE", "True", "95% CR", "Marginal 95% CI"),
       col = c("#8B0000", "#006400", "#2E4057", "#808080"),
       pch = c(19, 17, NA, NA),
       lty = c(NA, NA, 1, 3),
       lwd = c(NA, NA, 2, 1.5),
       bty = "n")
grid(col = "gray90")

# }

Run the code above in your browser using DataLab