Hessian Matrix of the Negative Log-Likelihood for the kkw Distribution

Computes the analytic 4x4 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the Kumaraswamy-Kumaraswamy (kkw) distribution with parameters alpha ($\alpha$), beta ($\beta$), delta ($\delta$), and lambda ($\lambda$). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\gamma = 1$. The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hskkw(par, data)

Arguments

par: A numeric vector of length 4 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), delta ($\delta \ge 0$), lambda ($\lambda > 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a 4x4 numeric matrix representing the Hessian matrix of the negative log-likelihood function, $-\partial^2 \ell / (\partial \theta_i \partial \theta_j)$, where $\theta = (\alpha, \beta, \delta, \lambda)$. Returns a 4x4 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).

Details

This function calculates the analytic second partial derivatives of the negative log-likelihood function based on the kkw log-likelihood ($\gamma=1$ case of GKw, see llkkw): $$ \ell(\theta | \mathbf{x}) = n[\ln(\delta+1) + \ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i) + \delta\ln(z_i)] $$ where $\theta = (\alpha, \beta, \delta, \lambda)$ 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 $- \frac{\partial^2 \ell(\theta | \mathbf{x})}{\partial \theta_i \partial \theta_j}$ for $\theta_i, \theta_j \in \{\alpha, \beta, \delta, \lambda\}$.

Key properties of the returned matrix:

Dimensions: 4x4.
Symmetry: The matrix is symmetric.
Ordering: Rows and columns correspond to the parameters in the order $\alpha, \beta, \delta, \lambda$.
Content: Analytic second derivatives of the negative log-likelihood.

This corresponds to the relevant submatrix of the 5x5 GKw Hessian (hsgkw) evaluated at $\gamma=1$. The exact analytical formulas are implemented directly.

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.

Author

Lopes, J. E.

Examples

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

# Generate sample data
set.seed(123)
n <- 1000
true_params <- c(alpha = 2.0, beta = 3.0, delta = 1.5, lambda = 2.0)
data <- rkkw(n, alpha = true_params[1], beta = true_params[2],
             delta = true_params[3], lambda = true_params[4])

# Evaluate Hessian at true parameters
hess_true <- hskkw(par = true_params, data = data)
cat("Hessian matrix at true parameters:\n")
#> Hessian matrix at true parameters:
print(hess_true, digits = 4)
#>        [,1]   [,2]   [,3]   [,4]
#> [1,] 1557.6 -564.6 -425.5  910.9
#> [2,] -564.6  247.6  195.1 -307.5
#> [3,] -425.5  195.1  160.0 -225.7
#> [4,]  910.9 -307.5 -225.7  541.1

# Check symmetry
cat("\nSymmetry check (max |H - H^T|):",
    max(abs(hess_true - t(hess_true))), "\n")
#> 
#> Symmetry check (max |H - H^T|): 0 


## Example 2: Hessian Properties at MLE

# Fit model
fit <- optim(
  par = c(1.5, 2.5, 1.0, 1.5),
  fn = llkkw,
  gr = grkkw,
  data = data,
  method = "BFGS",
  hessian = TRUE
)

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

# Hessian at MLE
hessian_at_mle <- hskkw(par = mle, data = data)
cat("\nHessian at MLE:\n")
#> 
#> Hessian at MLE:
print(hessian_at_mle, digits = 4)
#>        [,1]   [,2]   [,3]   [,4]
#> [1,] 1188.9 -371.1 -423.3  881.2
#> [2,] -371.1  140.8  168.0 -251.3
#> [3,] -423.3  168.0  202.5 -280.4
#> [4,]  881.2 -251.3 -280.4  678.3

# Compare with optim's numerical Hessian
cat("\nComparison with optim Hessian:\n")
#> 
#> Comparison with optim Hessian:
cat("Max absolute difference:",
    max(abs(hessian_at_mle - fit$hessian)), "\n")
#> Max absolute difference: 0.0003078667 

# Eigenvalue analysis
eigenvals <- eigen(hessian_at_mle, only.values = TRUE)$values
cat("\nEigenvalues:\n")
#> 
#> Eigenvalues:
print(eigenvals)
#> [1] 2.109731e+03 9.911840e+01 1.552331e+00 9.866428e-03

cat("\nPositive definite:", all(eigenvals > 0), "\n")
#> 
#> Positive definite: TRUE 
cat("Condition number:", max(eigenvals) / min(eigenvals), "\n")
#> Condition number: 213829.2 


## 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")
#> 
#> Variance-Covariance Matrix:
print(vcov_matrix, digits = 6)
#>          [,1]     [,2]      [,3]      [,4]
#> [1,]  4.72527  17.8381  -9.97973  -3.65563
#> [2,] 17.83805  71.2209 -40.54649 -13.54957
#> [3,] -9.97973 -40.5465  23.21460   7.54000
#> [4,] -3.65563 -13.5496   7.54000   2.84775

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

# Correlation matrix
corr_matrix <- cov2cor(vcov_matrix)
cat("\nCorrelation Matrix:\n")
#> 
#> Correlation Matrix:
print(corr_matrix, digits = 4)
#>         [,1]    [,2]    [,3]    [,4]
#> [1,]  1.0000  0.9724 -0.9529 -0.9965
#> [2,]  0.9724  1.0000 -0.9972 -0.9514
#> [3,] -0.9529 -0.9972  1.0000  0.9273
#> [4,] -0.9965 -0.9514  0.9273  1.0000

# Confidence intervals
z_crit <- qnorm(0.975)
results <- data.frame(
  Parameter = c("alpha", "beta", "delta", "lambda"),
  True = true_params,
  MLE = mle,
  SE = se,
  CI_Lower = mle - z_crit * se,
  CI_Upper = mle + z_crit * se
)
print(results, digits = 4)
#>        Parameter True   MLE    SE CI_Lower CI_Upper
#> alpha      alpha  2.0 2.304 2.174   -1.956    6.565
#> beta        beta  3.0 3.610 8.439  -12.931   20.150
#> delta      delta  1.5 1.222 4.818   -8.221   10.666
#> lambda    lambda  2.0 1.705 1.688   -1.603    5.012


## Example 4: Determinant and Trace Analysis

# Compute at different points
test_params <- rbind(
  c(1.5, 2.5, 1.0, 1.5),
  c(2.0, 3.0, 1.5, 2.0),
  mle,
  c(2.5, 3.5, 2.0, 2.5)
)

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

for (i in 1:nrow(test_params)) {
  H <- hskkw(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],
    Delta = test_params[i, 3],
    Lambda = test_params[i, 4],
    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")
#> 
#> Hessian Properties at Different Points:
print(hess_properties, digits = 4, row.names = FALSE)
#>  Alpha Beta Delta Lambda Determinant Trace Min_Eigenval Max_Eigenval
#>  1.500 2.50 1.000  1.500  -5.769e+09  3399   -1.056e+02         3061
#>  2.000 3.00 1.500  2.000  -1.317e+06  2506   -2.387e+00         2413
#>  2.304 3.61 1.222  1.705   3.203e+03  2210    9.866e-03         2110
#>  2.500 3.50 2.000  2.500  -2.204e+09  1873   -2.309e+02         1954
#>  Cond_Number
#>      -28.985
#>    -1011.104
#>   213829.226
#>       -8.462


## Example 5: Curvature Visualization (Alpha vs Beta)

# Create grid around MLE
alpha_grid <- seq(mle[1] - 1, mle[1] + 1, length.out = round(n/4))
beta_grid <- seq(mle[2] - 1, mle[2] + 1, length.out = round(n/4))
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 <- hskkw(c(alpha_grid[i], beta_grid[j], mle[3], mle[4]), 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: Confidence Ellipse (Alpha vs Beta)

# Extract 2x2 submatrix for alpha and beta
vcov_2d <- vcov_matrix[1:2, 1:2]

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

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

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

# Plot
plot(ellipse[, 1], ellipse[, 2], type = "l", lwd = 2, col = "#2E4057",
     xlab = expression(alpha), ylab = expression(beta),
     main = "95% Confidence Ellipse (Alpha vs Beta)", las = 1)

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



## Example 7: Confidence Ellipse (Delta vs Lambda)

# Extract 2x2 submatrix for delta and lambda
vcov_2d_dl <- vcov_matrix[3:4, 3:4]

# Create confidence ellipse
eig_decomp_dl <- eigen(vcov_2d_dl)
ellipse_dl <- matrix(NA, nrow = round(n/2), ncol = 2)
for (i in 1:round(n/2)) {
  v <- c(cos(theta[i]), sin(theta[i]))
  ellipse_dl[i, ] <- mle[3:4] + sqrt(chi2_val) *
    (eig_decomp_dl$vectors %*% diag(sqrt(eig_decomp_dl$values)) %*% v)
}

# Marginal confidence intervals
se_2d_dl <- sqrt(diag(vcov_2d_dl))
ci_delta <- mle[3] + c(-1, 1) * 1.96 * se_2d_dl[1]
ci_lambda <- mle[4] + c(-1, 1) * 1.96 * se_2d_dl[2]

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

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

points(mle[3], mle[4], pch = 19, col = "#8B0000", cex = 1.5)
points(true_params[3], true_params[4], 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")


# }