grmc: Gradient of the Negative Log-Likelihood for the McDonald...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Gradient of the Negative Log-Likelihood for the McDonald (Mc)/Beta Power Distribution

Description

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma (\gamma), delta (\delta), and lambda (\lambda). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where \alpha = 1 and \beta = 1. The gradient is useful for optimization.

Usage

grmc(par, data)

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).

Details

The components of the gradient vector of the negative log-likelihood (-\nabla \ell(\theta | \mathbf{x})) for the Mc (\alpha=1, \beta=1) model are:

-\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma+\delta+1) - \psi(\gamma)] - \lambda\sum_{i=1}^{n}\ln(x_i)

-\frac{\partial \ell}{\partial \delta} = n[\psi(\gamma+\delta+1) - \psi(\delta+1)] - \sum_{i=1}^{n}\ln(1-x_i^{\lambda})

-\frac{\partial \ell}{\partial \lambda} = -\frac{n}{\lambda} - \gamma\sum_{i=1}^{n}\ln(x_i) + \delta\sum_{i=1}^{n}\frac{x_i^{\lambda}\ln(x_i)}{1-x_i^{\lambda}}

where \psi(\cdot) is the digamma function (digamma). These formulas represent the derivatives of -\ell(\theta), consistent with minimizing the negative log-likelihood. They correspond to the relevant components of the general GKw gradient (grgkw) evaluated at \alpha=1, \beta=1.

Value

Returns a numeric vector of length 3 containing the partial derivatives of the negative log-likelihood function -\ell(\theta | \mathbf{x}) with respect to each parameter: (-\partial \ell/\partial \gamma, -\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).

Author(s)

Lopes, J. E.

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,

(Note: Specific gradient formulas might be derived or sourced from additional references).

See Also

grgkw (parent distribution gradient), llmc (negative log-likelihood for Mc), hsmc (Hessian for Mc, if available), dmc (density for Mc), optim, grad (for numerical gradient comparison), digamma.

Examples

# Assuming existence of rmc, llmc, grmc, hsmc functions for Mc distribution

# Generate sample data
set.seed(123)
true_par_mc <- c(gamma = 2, delta = 3, lambda = 0.5)
sample_data_mc <- rmc(100, gamma = true_par_mc[1], delta = true_par_mc[2],
                      lambda = true_par_mc[3])
hist(sample_data_mc, breaks = 20, main = "Mc(2, 3, 0.5) Sample")

# --- Find MLE estimates ---
start_par_mc <- c(1.5, 2.5, 0.8)
mle_result_mc <- stats::optim(par = start_par_mc,
                              fn = llmc,
                              gr = grmc, # Use analytical gradient for Mc
                              method = "BFGS",
                              hessian = TRUE,
                              data = sample_data_mc)

# --- Compare analytical gradient to numerical gradient ---
if (mle_result_mc$convergence == 0 &&
    requireNamespace("numDeriv", quietly = TRUE)) {

  mle_par_mc <- mle_result_mc$par
  cat("\nComparing Gradients for Mc at MLE estimates:\n")

  # Numerical gradient of llmc
  num_grad_mc <- numDeriv::grad(func = llmc, x = mle_par_mc, data = sample_data_mc)

  # Analytical gradient from grmc
  ana_grad_mc <- grmc(par = mle_par_mc, data = sample_data_mc)

  cat("Numerical Gradient (Mc):\n")
  print(num_grad_mc)
  cat("Analytical Gradient (Mc):\n")
  print(ana_grad_mc)

  # Check differences
  cat("Max absolute difference between Mc gradients:\n")
  print(max(abs(num_grad_mc - ana_grad_mc)))

} else {
  cat("\nSkipping Mc gradient comparison.\n")
}

# Example with Hessian comparison (if hsmc exists)
if (mle_result_mc$convergence == 0 &&
    requireNamespace("numDeriv", quietly = TRUE) && exists("hsmc")) {

  num_hess_mc <- numDeriv::hessian(func = llmc, x = mle_par_mc, data = sample_data_mc)
  ana_hess_mc <- hsmc(par = mle_par_mc, data = sample_data_mc)
  cat("\nMax absolute difference between Mc Hessians:\n")
  print(max(abs(num_hess_mc - ana_hess_mc)))

}

gkwreg documentation built on April 16, 2025, 1:10 a.m.