llmc: Negative Log-Likelihood for the McDonald (Mc)/Beta Power Distribution

Description

Computes the negative log-likelihood function for the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma ($\gamma$), delta ($\delta$), and lambda ($\lambda$), given a vector of observations. This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\alpha = 1$ and $\beta = 1$. This function is suitable for maximum likelihood estimation.

Usage

llmc(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 3 containing the distribution parameters in the order: gamma ($\gamma > 0$), 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 McDonald (Mc) distribution is the GKw distribution (dmc) with $\alpha=1$ and $\beta=1$. Its probability density function (PDF) is: $$ f(x | \theta) = \frac{\lambda}{B(\gamma,\delta+1)} x^{\gamma \lambda - 1} (1 - x^\lambda)^\delta $$ for $0 < x < 1$, $\theta = (\gamma, \delta, \lambda)$, and $B(a,b)$ is the Beta function (beta). The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = n[\ln(\lambda) - \ln B(\gamma, \delta+1)] + \sum_{i=1}^{n} [(\gamma\lambda - 1)\ln(x_i) + \delta\ln(1 - x_i^\lambda)] $$ This function computes and returns the negative log-likelihood, $-\ell(\theta|\mathbf{x})$, suitable for minimization using optimization routines like optim. Numerical stability is maintained, including using the log-gamma function (lgamma) for the Beta function term.

References

McDonald, J. B. (1984). Some generalized functions for the size distribution of income. Econometrica, 52(3), 647-663.

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 Log-Likelihood Evaluation

# Generate sample data with more stable parameters
set.seed(123)
n <- 1000
true_params <- c(gamma = 2.0, delta = 2.5, lambda = 1.5)
data <- rmc(n, gamma = true_params[1], delta = true_params[2],
            lambda = true_params[3])

# Evaluate negative log-likelihood at true parameters
nll_true <- llmc(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, 2.0, 1.0),
  c(2.0, 2.5, 1.5),
  c(2.5, 3.0, 2.0)
)

nll_values <- apply(test_params, 1, function(p) llmc(p, data))
results <- data.frame(
  Gamma = test_params[, 1],
  Delta = test_params[, 2],
  Lambda = test_params[, 3],
  NegLogLik = nll_values
)
print(results, digits = 4)


## Example 2: Maximum Likelihood Estimation

# Optimization using BFGS with analytical gradient
fit <- optim(
  par = c(1.5, 2.0, 1.0),
  fn = llmc,
  gr = grmc,
  data = data,
  method = "BFGS",
  hessian = TRUE
)

mle <- fit$par
names(mle) <- c("gamma", "delta", "lambda")
se <- sqrt(diag(solve(fit$hessian)))

results <- data.frame(
  Parameter = c("gamma", "delta", "lambda"),
  True = true_params,
  MLE = mle,
  SE = se,
  CI_Lower = mle - 1.96 * se,
  CI_Upper = mle + 1.96 * se
)
print(results, digits = 4)

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, 2.0, 1.0)

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

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

  comparison <- rbind(comparison, data.frame(
    Method = method,
    Gamma = fit_temp$par[1],
    Delta = fit_temp$par[2],
    Lambda = fit_temp$par[3],
    NegLogLik = fit_temp$value,
    Convergence = fit_temp$convergence,
    stringsAsFactors = FALSE
  ))
}

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


## Example 4: Likelihood Ratio Test

# Test H0: lambda = 1.5 vs H1: lambda free
loglik_full <- -fit$value

restricted_ll <- function(params_restricted, data, lambda_fixed) {
  llmc(par = c(params_restricted[1], params_restricted[2],
               lambda_fixed), data = data)
}

fit_restricted <- optim(
  par = c(mle[1], mle[2]),
  fn = restricted_ll,
  data = data,
  lambda_fixed = 1.5,
  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[-1],
    fn = function(p) llmc(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[-2],
    fn = function(p) llmc(c(p[1], delta_grid[i], p[2]), data),
    method = "BFGS"
  )
  profile_ll_delta[i] <- -profile_fit$value
}

# Profile for lambda
lambda_grid <- seq(mle[3] - 1.5, mle[3] + 1.5, length.out = 50)
lambda_grid <- lambda_grid[lambda_grid > 0]
profile_ll_lambda <- numeric(length(lambda_grid))

for (i in seq_along(lambda_grid)) {
  profile_fit <- optim(
    par = mle[-3],
    fn = function(p) llmc(c(p[1], p[2], lambda_grid[i]), data),
    method = "BFGS"
  )
  profile_ll_lambda[i] <- -profile_fit$value
}

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

# Plot all profiles

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")

plot(lambda_grid, profile_ll_lambda, type = "l", lwd = 2, col = "#2E4057",
     xlab = expression(lambda), ylab = "Profile Log-Likelihood",
     main = expression(paste("Profile: ", lambda)), las = 1)
abline(v = mle[3], col = "#8B0000", lty = 2, lwd = 2)
abline(v = true_params[3], 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 Surfaces (All pairs side by side)

# Create 2D grids 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))
lambda_2d <- seq(mle[3] - 1.5, mle[3] + 1.5, length.out = round(n/25))

gamma_2d <- gamma_2d[gamma_2d > 0]
delta_2d <- delta_2d[delta_2d > 0]
lambda_2d <- lambda_2d[lambda_2d > 0]

# Compute all log-likelihood surfaces
ll_surface_gd <- matrix(NA, nrow = length(gamma_2d), ncol = length(delta_2d))
ll_surface_gl <- matrix(NA, nrow = length(gamma_2d), ncol = length(lambda_2d))
ll_surface_dl <- matrix(NA, nrow = length(delta_2d), ncol = length(lambda_2d))

# Gamma vs Delta
for (i in seq_along(gamma_2d)) {
  for (j in seq_along(delta_2d)) {
    ll_surface_gd[i, j] <- -llmc(c(gamma_2d[i], delta_2d[j], mle[3]), data)
  }
}

# Gamma vs Lambda
for (i in seq_along(gamma_2d)) {
  for (j in seq_along(lambda_2d)) {
    ll_surface_gl[i, j] <- -llmc(c(gamma_2d[i], mle[2], lambda_2d[j]), data)
  }
}

# Delta vs Lambda
for (i in seq_along(delta_2d)) {
  for (j in seq_along(lambda_2d)) {
    ll_surface_dl[i, j] <- -llmc(c(mle[1], delta_2d[i], lambda_2d[j]), data)
  }
}

# Confidence region levels
max_ll_gd <- max(ll_surface_gd, na.rm = TRUE)
max_ll_gl <- max(ll_surface_gl, na.rm = TRUE)
max_ll_dl <- max(ll_surface_dl, na.rm = TRUE)

levels_95_gd <- max_ll_gd - qchisq(0.95, df = 2) / 2
levels_95_gl <- max_ll_gl - qchisq(0.95, df = 2) / 2
levels_95_dl <- max_ll_dl - qchisq(0.95, df = 2) / 2

# Plot 

# Gamma vs Delta
contour(gamma_2d, delta_2d, ll_surface_gd,
        xlab = expression(gamma), ylab = expression(delta),
        main = "Gamma vs Delta", las = 1,
        levels = seq(min(ll_surface_gd, na.rm = TRUE), max_ll_gd, length.out = 20),
        col = "#2E4057", lwd = 1)
contour(gamma_2d, delta_2d, ll_surface_gd,
        levels = levels_95_gd, col = "#FF6347", lwd = 2.5, lty = 1, add = TRUE)
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")

# Gamma vs Lambda
contour(gamma_2d, lambda_2d, ll_surface_gl,
        xlab = expression(gamma), ylab = expression(lambda),
        main = "Gamma vs Lambda", las = 1,
        levels = seq(min(ll_surface_gl, na.rm = TRUE), max_ll_gl, length.out = 20),
        col = "#2E4057", lwd = 1)
contour(gamma_2d, lambda_2d, ll_surface_gl,
        levels = levels_95_gl, col = "#FF6347", lwd = 2.5, lty = 1, add = TRUE)
points(mle[1], mle[3], pch = 19, col = "#8B0000", cex = 1.5)
points(true_params[1], true_params[3], pch = 17, col = "#006400", cex = 1.5)
grid(col = "gray90")

# Delta vs Lambda
contour(delta_2d, lambda_2d, ll_surface_dl,
        xlab = expression(delta), ylab = expression(lambda),
        main = "Delta vs Lambda", las = 1,
        levels = seq(min(ll_surface_dl, na.rm = TRUE), max_ll_dl, length.out = 20),
        col = "#2E4057", lwd = 1)
contour(delta_2d, lambda_2d, ll_surface_dl,
        levels = levels_95_dl, col = "#FF6347", lwd = 2.5, lty = 1, add = TRUE)
points(mle[2], mle[3], pch = 19, col = "#8B0000", cex = 1.5)
points(true_params[2], true_params[3], pch = 17, col = "#006400", cex = 1.5)
grid(col = "gray90")

legend("topright",
       legend = c("MLE", "True", "95% CR"),
       col = c("#8B0000", "#006400", "#FF6347"),
       pch = c(19, 17, NA),
       lty = c(NA, NA, 1),
       lwd = c(NA, NA, 2.5),
       bty = "n", cex = 0.8)

# }

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