gkwreg (version 1.0.7)

grkw: 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)

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).

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).

Author

Lopes, J. E.

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 \(\gamma=1, \delta=0, \lambda=1\).

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

# \donttest{
# 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)))

}

# }