grkw: Gradient of the Negative Log-Likelihood for the Kumaraswamy...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Gradient of the Negative Log-Likelihood for the Kumaraswamy (Kw) Distribution

Description

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the two-parameter Kumaraswamy (Kw) distribution with parameters alpha (\alpha) and beta (\beta). This provides the analytical gradient often used for efficient optimization via maximum likelihood estimation.

Usage

grkw(par, data)

Arguments

par	A numeric vector of length 2 containing the distribution parameters in the order: alpha (\alpha > 0), beta (\beta > 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 Kw 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}

-\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - \sum_{i=1}^{n}\ln(v_i)

where v_i = 1 - x_i^{\alpha}. 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 \gamma=1, \delta=0, \lambda=1.

Value

Returns a numeric vector of length 2 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). 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

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

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

See Also

grgkw (parent distribution gradient), llkw (negative log-likelihood for Kw), hskw (Hessian for Kw, if available), dkw (density for Kw), optim, grad (for numerical gradient comparison).

Examples

# Assuming existence of rkw, llkw, grkw, hskw functions for Kw

# Generate sample data
set.seed(123)
true_par_kw <- c(alpha = 2, beta = 3)
sample_data_kw <- rkw(100, alpha = true_par_kw[1], beta = true_par_kw[2])
hist(sample_data_kw, breaks = 20, main = "Kw(2, 3) Sample")

# --- Find MLE estimates ---
start_par_kw <- c(1.5, 2.5)
mle_result_kw <- stats::optim(par = start_par_kw,
                              fn = llkw,
                              gr = grkw, # Use analytical gradient for Kw
                              method = "L-BFGS-B", # Recommended for bounds
                              lower = c(1e-6, 1e-6),
                              hessian = TRUE,
                              data = sample_data_kw)

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

  mle_par_kw <- mle_result_kw$par
  cat("\nComparing Gradients for Kw at MLE estimates:\n")

  # Numerical gradient of llkw
  num_grad_kw <- numDeriv::grad(func = llkw, x = mle_par_kw, data = sample_data_kw)

  # Analytical gradient from grkw
  ana_grad_kw <- grkw(par = mle_par_kw, data = sample_data_kw)

  cat("Numerical Gradient (Kw):\n")
  print(num_grad_kw)
  cat("Analytical Gradient (Kw):\n")
  print(ana_grad_kw)

  # Check differences
  cat("Max absolute difference between Kw gradients:\n")
  print(max(abs(num_grad_kw - ana_grad_kw)))

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

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

  num_hess_kw <- numDeriv::hessian(func = llkw, x = mle_par_kw, data = sample_data_kw)
  ana_hess_kw <- hskw(par = mle_par_kw, data = sample_data_kw)
  cat("\nMax absolute difference between Kw Hessians:\n")
  print(max(abs(num_hess_kw - ana_hess_kw)))

}

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