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).

Examples

Run this code

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

# Generate sample data
set.seed(123)
n <- 1000
true_params <- c(alpha = 2.5, beta = 3.5)
data <- rkw(n, alpha = true_params[1], beta = true_params[2])

# Evaluate Hessian at true parameters
hess_true <- hskw(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(2, 2),
  fn = llkw,
  gr = grkw,
  data = data,
  method = "BFGS",
  hessian = TRUE
)

mle <- fit$par
names(mle) <- c("alpha", "beta")

# Hessian at MLE
hessian_at_mle <- hskw(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 (negative Hessian for neg-loglik)
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("alpha", "beta")

# 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("alpha", "beta"),
  True = true_params,
  MLE = mle,
  SE = se,
  CI_Lower = mle - z_crit * se,
  CI_Upper = mle + z_crit * se
)
print(results, digits = 4)

## 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(3.0, 4.0)
)

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

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

  hess_properties <- rbind(hess_properties, data.frame(
    Alpha = test_params[i, 1],
    Beta = 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

# Create grid around MLE
alpha_grid <- seq(mle[1] - 0.5, mle[1] + 0.5, length.out = 30)
beta_grid <- seq(mle[2] - 0.5, mle[2] + 0.5, length.out = 30)
alpha_grid <- alpha_grid[alpha_grid > 0]
beta_grid <- beta_grid[beta_grid > 0]

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

for (i in seq_along(alpha_grid)) {
  for (j in seq_along(beta_grid)) {
    H <- hskw(c(alpha_grid[i], beta_grid[j]), data)
    determinant_surface[i, j] <- det(H)
    trace_surface[i, j] <- sum(diag(H))
  }
}

# Plot

contour(alpha_grid, beta_grid, determinant_surface,
        xlab = expression(alpha), ylab = expression(beta),
        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(alpha_grid, beta_grid, trace_surface,
        xlab = expression(alpha), ylab = expression(beta),
        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: Fisher Information and Asymptotic Efficiency

# Observed information (at MLE)
obs_fisher <- hessian_at_mle

# Asymptotic covariance matrix
asymp_cov <- solve(obs_fisher)

cat("\nAsymptotic Standard Errors:\n")
cat("SE(alpha):", sqrt(asymp_cov[1, 1]), "\n")
cat("SE(beta):", sqrt(asymp_cov[2, 2]), "\n")

# Cramér-Rao Lower Bound
cat("\nCramér-Rao Lower Bounds:\n")
cat("CRLB(alpha):", sqrt(asymp_cov[1, 1]), "\n")
cat("CRLB(beta):", sqrt(asymp_cov[2, 2]), "\n")

# Efficiency ellipse (95% confidence region)
theta <- seq(0, 2 * pi, length.out = 100)
chi2_val <- qchisq(0.95, df = 2)

# Eigendecomposition
eig_decomp <- eigen(asymp_cov)

# Ellipse points
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)
}

# Plot confidence ellipse

plot(ellipse[, 1], ellipse[, 2], type = "l", lwd = 2, col = "#2E4057",
     xlab = expression(alpha), ylab = expression(beta),
     main = "95% Confidence Ellipse", las = 1)
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"),
       col = c("#8B0000", "#006400", "#2E4057"),
       pch = c(19, 17, NA), lty = c(NA, NA, 1),
       lwd = c(NA, NA, 2), bty = "n")
grid(col = "gray90")

# }

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