llbeta: Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization)

Description

Computes 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. This function is suitable for maximum likelihood estimation.

Usage

llbeta(par, data)

Value

Returns a single double value representing the negative log-likelihood ($-\ell(\theta|\mathbf{x})$). Returns Inf

if any parameter values in par 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 negative log-likelihood for a Beta distribution with parameters shape1 = gamma ($\gamma$) and shape2 = delta + 1 ($\delta+1$). The probability density function (PDF) is: $$ f(x | \gamma, \delta) = \frac{x^{\gamma-1} (1-x)^{\delta}}{B(\gamma, \delta+1)} $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (beta). The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = \sum_{i=1}^{n} [(\gamma-1)\ln(x_i) + \delta\ln(1-x_i)] - n \ln B(\gamma, \delta+1) $$ where $\theta = (\gamma, \delta)$. This function computes and returns the negative log-likelihood, $-\ell(\theta|\mathbf{x})$, suitable for minimization using optimization routines like optim. It is equivalent to the negative log-likelihood of the GKw distribution (llgkw) evaluated at $\alpha=1, \beta=1, \lambda=1$, and also to the negative log-likelihood of the McDonald distribution (llmc) evaluated at $\lambda=1$. The term $\ln B(\gamma, \delta+1)$ is typically computed using log-gamma functions (lgamma) for numerical stability.

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,

Examples

Run this code

# \donttest{
## Example 1: Basic Log-Likelihood 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 negative log-likelihood at true parameters
nll_true <- llbeta(par = true_params, data = data)
cat("Negative log-likelihood at true parameters:", nll_true, "\n")

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

nll_values <- apply(test_params, 1, function(p) llbeta(p, data))
results <- data.frame(
  Gamma = test_params[, 1],
  Delta = test_params[, 2],
  NegLogLik = nll_values
)
print(results, digits = 4)


## Example 2: Maximum Likelihood Estimation

# Optimization using L-BFGS-B with bounds
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$par
names(mle) <- c("gamma", "delta")
se <- sqrt(diag(solve(fit$hessian)))

results <- data.frame(
  Parameter = c("gamma", "delta"),
  True = true_params,
  MLE = mle,
  SE = se,
  CI_Lower = mle - 1.96 * se,
  CI_Upper = mle + 1.96 * 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)

cat("\nNegative log-likelihood at MLE:", fit$value, "\n")
cat("AIC:", 2 * fit$value + 2 * length(mle), "\n")
cat("BIC:", 2 * fit$value + length(mle) * log(n), "\n")


## Example 3: Comparing Optimization Methods

methods <- c("BFGS", "L-BFGS-B", "Nelder-Mead", "CG")
start_params <- c(1.5, 2.5)

comparison <- data.frame(
  Method = character(),
  Gamma = numeric(),
  Delta = numeric(),
  NegLogLik = numeric(),
  Convergence = integer(),
  stringsAsFactors = FALSE
)

for (method in methods) {
  if (method %in% c("BFGS", "CG")) {
    fit_temp <- optim(
      par = start_params,
      fn = llbeta,
      gr = grbeta,
      data = data,
      method = method
    )
  } else if (method == "L-BFGS-B") {
    fit_temp <- optim(
      par = start_params,
      fn = llbeta,
      gr = grbeta,
      data = data,
      method = method,
      lower = c(0.01, 0.01),
      upper = c(100, 100)
    )
  } else {
    fit_temp <- optim(
      par = start_params,
      fn = llbeta,
      data = data,
      method = method
    )
  }

  comparison <- rbind(comparison, data.frame(
    Method = method,
    Gamma = fit_temp$par[1],
    Delta = fit_temp$par[2],
    NegLogLik = fit_temp$value,
    Convergence = fit_temp$convergence,
    stringsAsFactors = FALSE
  ))
}

print(comparison, digits = 4, row.names = FALSE)


## Example 4: Likelihood Ratio Test

# Test H0: delta = 3 vs H1: delta free
loglik_full <- -fit$value

restricted_ll <- function(params_restricted, data, delta_fixed) {
  llbeta(par = c(params_restricted[1], delta_fixed), data = data)
}

fit_restricted <- optim(
  par = mle[1],
  fn = restricted_ll,
  data = data,
  delta_fixed = 3,
  method = "BFGS"
)

loglik_restricted <- -fit_restricted$value
lr_stat <- 2 * (loglik_full - loglik_restricted)
p_value <- pchisq(lr_stat, df = 1, lower.tail = FALSE)

cat("LR Statistic:", round(lr_stat, 4), "\n")
cat("P-value:", format.pval(p_value, digits = 4), "\n")


## Example 5: Univariate Profile Likelihoods

# Profile for gamma
gamma_grid <- seq(mle[1] - 1.5, mle[1] + 1.5, length.out = 50)
gamma_grid <- gamma_grid[gamma_grid > 0]
profile_ll_gamma <- numeric(length(gamma_grid))

for (i in seq_along(gamma_grid)) {
  profile_fit <- optim(
    par = mle[2],
    fn = function(p) llbeta(c(gamma_grid[i], p), data),
    method = "BFGS"
  )
  profile_ll_gamma[i] <- -profile_fit$value
}

# Profile for delta
delta_grid <- seq(mle[2] - 1.5, mle[2] + 1.5, length.out = 50)
delta_grid <- delta_grid[delta_grid > 0]
profile_ll_delta <- numeric(length(delta_grid))

for (i in seq_along(delta_grid)) {
  profile_fit <- optim(
    par = mle[1],
    fn = function(p) llbeta(c(p, delta_grid[i]), data),
    method = "BFGS"
  )
  profile_ll_delta[i] <- -profile_fit$value
}

# 95% confidence threshold
chi_crit <- qchisq(0.95, df = 1)
threshold <- max(profile_ll_gamma) - chi_crit / 2

# Plot 

plot(gamma_grid, profile_ll_gamma, type = "l", lwd = 2, col = "#2E4057",
     xlab = expression(gamma), ylab = "Profile Log-Likelihood",
     main = expression(paste("Profile: ", gamma)), las = 1)
abline(v = mle[1], col = "#8B0000", lty = 2, lwd = 2)
abline(v = true_params[1], col = "#006400", lty = 2, lwd = 2)
abline(h = threshold, col = "#808080", lty = 3, lwd = 1.5)
legend("topright", legend = c("MLE", "True", "95% CI"),
       col = c("#8B0000", "#006400", "#808080"),
       lty = c(2, 2, 3), lwd = 2, bty = "n", cex = 0.8)
grid(col = "gray90")

plot(delta_grid, profile_ll_delta, type = "l", lwd = 2, col = "#2E4057",
     xlab = expression(delta), ylab = "Profile Log-Likelihood",
     main = expression(paste("Profile: ", delta)), las = 1)
abline(v = mle[2], col = "#8B0000", lty = 2, lwd = 2)
abline(v = true_params[2], col = "#006400", lty = 2, lwd = 2)
abline(h = threshold, col = "#808080", lty = 3, lwd = 1.5)
legend("topright", legend = c("MLE", "True", "95% CI"),
       col = c("#8B0000", "#006400", "#808080"),
       lty = c(2, 2, 3), lwd = 2, bty = "n", cex = 0.8)
grid(col = "gray90")


## Example 6: 2D Log-Likelihood Surface (Gamma vs Delta)

# Create 2D grid with wider range (±1.5)
gamma_2d <- seq(mle[1] - 1.5, mle[1] + 1.5, length.out = round(n/25))
delta_2d <- seq(mle[2] - 1.5, mle[2] + 1.5, length.out = round(n/25))
gamma_2d <- gamma_2d[gamma_2d > 0]
delta_2d <- delta_2d[delta_2d > 0]

# Compute log-likelihood surface
ll_surface_gd <- matrix(NA, nrow = length(gamma_2d), ncol = length(delta_2d))

for (i in seq_along(gamma_2d)) {
  for (j in seq_along(delta_2d)) {
    ll_surface_gd[i, j] <- -llbeta(c(gamma_2d[i], delta_2d[j]), data)
  }
}

# Confidence region levels
max_ll_gd <- max(ll_surface_gd, na.rm = TRUE)
levels_90_gd <- max_ll_gd - qchisq(0.90, df = 2) / 2
levels_95_gd <- max_ll_gd - qchisq(0.95, df = 2) / 2
levels_99_gd <- max_ll_gd - qchisq(0.99, df = 2) / 2

# Plot contour

contour(gamma_2d, delta_2d, ll_surface_gd,
        xlab = expression(gamma), ylab = expression(delta),
        main = "2D Log-Likelihood: Gamma vs Delta",
        levels = seq(min(ll_surface_gd, na.rm = TRUE), max_ll_gd, length.out = 20),
        col = "#2E4057", las = 1, lwd = 1)

contour(gamma_2d, delta_2d, ll_surface_gd,
        levels = c(levels_90_gd, levels_95_gd, levels_99_gd),
        col = c("#FFA07A", "#FF6347", "#8B0000"),
        lwd = c(2, 2.5, 3), lty = c(3, 2, 1),
        add = TRUE, labcex = 0.8)

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", "90% CR", "95% CR", "99% CR"),
       col = c("#8B0000", "#006400", "#FFA07A", "#FF6347", "#8B0000"),
       pch = c(19, 17, NA, NA, NA),
       lty = c(NA, NA, 3, 2, 1),
       lwd = c(NA, NA, 2, 2.5, 3),
       bty = "n", cex = 0.8)
grid(col = "gray90")

# }

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