grkkw: 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)

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).

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).

Author

Lopes, J. E.

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.

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.

Examples

Run this code

# \donttest{
# 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
}

# }

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