grkkw: Gradient of the Negative Log-Likelihood for the kkw Distribution

Description

Computes the gradient vector (vector of first 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 gradient is typically used in optimization algorithms for maximum likelihood estimation.

Usage

grkkw(par, data)

Value

Returns a numeric vector of length 4 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \alpha, -\partial \ell/\partial \beta, -\partial \ell/\partial \delta, -\partial \ell/\partial \lambda)$. Returns a vector of 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 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).

Author

Lopes, J. E.

Details

The components of the gradient vector of the negative log-likelihood ($-\nabla \ell(\theta | \mathbf{x})$) for the kkw ($\gamma=1$) model are:

$$ -\frac{\partial \ell}{\partial \alpha} = -\frac{n}{\alpha} - \sum_{i=1}^{n}\ln(x_i) + (\beta-1)\sum_{i=1}^{n}\frac{x_i^{\alpha}\ln(x_i)}{v_i} - (\lambda-1)\sum_{i=1}^{n}\frac{\beta v_i^{\beta-1} x_i^{\alpha}\ln(x_i)}{w_i} + \delta\sum_{i=1}^{n}\frac{\lambda w_i^{\lambda-1} \beta v_i^{\beta-1} x_i^{\alpha}\ln(x_i)}{z_i} $$ $$ -\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - \sum_{i=1}^{n}\ln(v_i) + (\lambda-1)\sum_{i=1}^{n}\frac{v_i^{\beta}\ln(v_i)}{w_i} - \delta\sum_{i=1}^{n}\frac{\lambda w_i^{\lambda-1} v_i^{\beta}\ln(v_i)}{z_i} $$ $$ -\frac{\partial \ell}{\partial \delta} = -\frac{n}{\delta+1} - \sum_{i=1}^{n}\ln(z_i) $$ $$ -\frac{\partial \ell}{\partial \lambda} = -\frac{n}{\lambda} - \sum_{i=1}^{n}\ln(w_i) + \delta\sum_{i=1}^{n}\frac{w_i^{\lambda}\ln(w_i)}{z_i} $$

where:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$
$z_i = 1 - w_i^{\lambda} = 1 - [1-(1-x_i^{\alpha})^{\beta}]^{\lambda}$

These formulas represent the derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the general GKw gradient (grgkw) components for $\alpha, \beta, \delta, \lambda$ evaluated at $\gamma=1$. Note that the component for $\gamma$ is omitted. Numerical stability is maintained through careful implementation.

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.

Examples

Run this code

# \donttest{
## Example 1: Basic Gradient 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 gradient at true parameters
grad_true <- grkkw(par = true_params, data = data)
cat("Gradient at true parameters:\n")
print(grad_true)
cat("Norm:", sqrt(sum(grad_true^2)), "\n")

# Evaluate at different parameter values
test_params <- rbind(
  c(1.5, 2.5, 1.0, 1.5),
  c(2.0, 3.0, 1.5, 2.0),
  c(2.5, 3.5, 2.0, 2.5)
)

grad_norms <- apply(test_params, 1, function(p) {
  g <- grkkw(p, data)
  sqrt(sum(g^2))
})

results <- data.frame(
  Alpha = test_params[, 1],
  Beta = test_params[, 2],
  Delta = test_params[, 3],
  Lambda = test_params[, 4],
  Grad_Norm = grad_norms
)
print(results, digits = 4)


## Example 2: Gradient in Optimization

# Optimization with analytical gradient
fit_with_grad <- optim(
  par = c(1.5, 2.5, 1.0, 1.5),
  fn = llkkw,
  gr = grkkw,
  data = data,
  method = "BFGS",
  hessian = TRUE,
  control = list(trace = 0)
)

# Optimization without gradient
fit_no_grad <- optim(
  par = c(1.5, 2.5, 1.0, 1.5),
  fn = llkkw,
  data = data,
  method = "BFGS",
  hessian = TRUE,
  control = list(trace = 0)
)

comparison <- data.frame(
  Method = c("With Gradient", "Without Gradient"),
  Alpha = c(fit_with_grad$par[1], fit_no_grad$par[1]),
  Beta = c(fit_with_grad$par[2], fit_no_grad$par[2]),
  Delta = c(fit_with_grad$par[3], fit_no_grad$par[3]),
  Lambda = c(fit_with_grad$par[4], fit_no_grad$par[4]),
  NegLogLik = c(fit_with_grad$value, fit_no_grad$value),
  Iterations = c(fit_with_grad$counts[1], fit_no_grad$counts[1])
)
print(comparison, digits = 4, row.names = FALSE)


## Example 3: Verifying Gradient at MLE

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

# At MLE, gradient should be approximately zero
gradient_at_mle <- grkkw(par = mle, data = data)
cat("\nGradient at MLE:\n")
print(gradient_at_mle)
cat("Max absolute component:", max(abs(gradient_at_mle)), "\n")
cat("Gradient norm:", sqrt(sum(gradient_at_mle^2)), "\n")


## Example 4: Numerical vs Analytical Gradient

# Manual finite difference gradient
numerical_gradient <- function(f, x, data, h = 1e-7) {
  grad <- numeric(length(x))
  for (i in seq_along(x)) {
    x_plus <- x_minus <- x
    x_plus[i] <- x[i] + h
    x_minus[i] <- x[i] - h
    grad[i] <- (f(x_plus, data) - f(x_minus, data)) / (2 * h)
  }
  return(grad)
}

# Compare at MLE
grad_analytical <- grkkw(par = mle, data = data)
grad_numerical <- numerical_gradient(llkkw, mle, data)

comparison_grad <- data.frame(
  Parameter = c("alpha", "beta", "delta", "lambda"),
  Analytical = grad_analytical,
  Numerical = grad_numerical,
  Abs_Diff = abs(grad_analytical - grad_numerical),
  Rel_Error = abs(grad_analytical - grad_numerical) /
              (abs(grad_analytical) + 1e-10)
)
print(comparison_grad, digits = 8)


## Example 5: Score Test Statistic

# Score test for H0: theta = theta0
theta0 <- c(1.8, 2.8, 1.3, 1.8)
score_theta0 <- -grkkw(par = theta0, data = data)

# Fisher information at theta0
fisher_info <- hskkw(par = theta0, data = data)

# Score test statistic
score_stat <- t(score_theta0) %*% solve(fisher_info) %*% score_theta0
p_value <- pchisq(score_stat, df = 4, lower.tail = FALSE)

cat("\nScore Test:\n")
cat("H0: alpha=1.8, beta=2.8, delta=1.3, lambda=1.8\n")
cat("Test statistic:", score_stat, "\n")
cat("P-value:", format.pval(p_value, digits = 4), "\n")


## Example 6: Confidence Ellipse with Gradient Information

# For visualization, use first two parameters (alpha, beta)
# Observed information
obs_info <- hskkw(par = mle, data = data)
vcov_full <- solve(obs_info)
vcov_2d <- vcov_full[1:2, 1:2]

# 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[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 Region (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")
# }

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