grkkw: Gradient of the Negative Log-Likelihood for the kkw...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Gradient of the Negative Log-Likelihood for the kkw Distribution

Description

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the Kumaraswamy-Kumaraswamy (kkw) distribution with parameters alpha (\alpha), beta (\beta), delta (\delta), and lambda (\lambda). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where \gamma = 1. The gradient is typically used in optimization algorithms for maximum likelihood estimation.

Usage

grkkw(par, data)

Arguments

par	A numeric vector of length 4 containing the distribution parameters in the order: alpha (\alpha > 0), beta (\beta > 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 kkw (\gamma=1) 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} - (\lambda-1)\sum_{i=1}^{n}\frac{\beta v_i^{\beta-1} x_i^{\alpha}\ln(x_i)}{w_i} + \delta\sum_{i=1}^{n}\frac{\lambda w_i^{\lambda-1} \beta v_i^{\beta-1} x_i^{\alpha}\ln(x_i)}{z_i}

-\frac{\partial \ell}{\partial \beta} = -\frac{n}{\beta} - \sum_{i=1}^{n}\ln(v_i) + (\lambda-1)\sum_{i=1}^{n}\frac{v_i^{\beta}\ln(v_i)}{w_i} - \delta\sum_{i=1}^{n}\frac{\lambda w_i^{\lambda-1} v_i^{\beta}\ln(v_i)}{z_i}

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

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

where:

• v_i = 1 - x_i^{\alpha}

• w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}

• z_i = 1 - w_i^{\lambda} = 1 - [1-(1-x_i^{\alpha})^{\beta}]^{\lambda}

These formulas represent the derivatives of -\ell(\theta), consistent with minimizing the negative log-likelihood. They correspond to the general GKw gradient (grgkw) components for \alpha, \beta, \delta, \lambda evaluated at \gamma=1. Note that the component for \gamma is omitted. Numerical stability is maintained through careful implementation.

Value

Returns a numeric vector of length 4 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, -\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

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.

See Also

grgkw (parent distribution gradient), llkkw (negative log-likelihood for kkw), hskkw (Hessian for kkw), dkkw (density for kkw), optim, grad (for numerical gradient comparison).

Examples

# Assuming existence of rkkw, llkkw, grkkw, hskkw functions for kkw

# Generate sample data
set.seed(123)
true_par_kkw <- c(alpha = 2, beta = 3, delta = 1.5, lambda = 0.5)
if (exists("rkkw")) {
  sample_data_kkw <- rkkw(100, alpha = true_par_kkw[1], beta = true_par_kkw[2],
                         delta = true_par_kkw[3], lambda = true_par_kkw[4])
} else {
  sample_data_kkw <- rgkw(100, alpha = true_par_kkw[1], beta = true_par_kkw[2],
                         gamma = 1, delta = true_par_kkw[3], lambda = true_par_kkw[4])
}

# --- Find MLE estimates ---
start_par_kkw <- c(1.5, 2.5, 1.0, 0.6)
mle_result_kkw <- stats::optim(par = start_par_kkw,
                               fn = llkkw,
                               gr = grkkw, # Use analytical gradient for kkw
                               method = "BFGS",
                               hessian = TRUE,
                               data = sample_data_kkw)

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

  mle_par_kkw <- mle_result_kkw$par
  cat("\nComparing Gradients for kkw at MLE estimates:\n")

  # Numerical gradient of llkkw
  num_grad_kkw <- numDeriv::grad(func = llkkw, x = mle_par_kkw, data = sample_data_kkw)

  # Analytical gradient from grkkw
  ana_grad_kkw <- grkkw(par = mle_par_kkw, data = sample_data_kkw)

  cat("Numerical Gradient (kkw):\n")
  print(num_grad_kkw)
  cat("Analytical Gradient (kkw):\n")
  print(ana_grad_kkw)

  # Check differences
  cat("Max absolute difference between kkw gradients:\n")
  print(max(abs(num_grad_kkw - ana_grad_kkw)))

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

# --- Optional: Compare with relevant components of GKw gradient ---
# Requires grgkw function
if (mle_result_kkw$convergence == 0 && exists("grgkw")) {
  # Create 5-param vector for grgkw (insert gamma=1)
  mle_par_gkw_equiv <- c(mle_par_kkw[1:2], gamma = 1.0, mle_par_kkw[3:4])
  ana_grad_gkw <- grgkw(par = mle_par_gkw_equiv, data = sample_data_kkw)
  # Extract components corresponding to alpha, beta, delta, lambda
  ana_grad_gkw_subset <- ana_grad_gkw[c(1, 2, 4, 5)]

  cat("\nComparison with relevant components of GKw gradient:\n")
  cat("Max absolute difference:\n")
  print(max(abs(ana_grad_kkw - ana_grad_gkw_subset))) # Should be very small
}

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