gkwreg (version 1.0.7)

llekw: Negative Log-Likelihood for the Exponentiated Kumaraswamy (EKw) Distribution

Description

Computes the negative log-likelihood function for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha (\(\alpha\)), beta (\(\beta\)), and lambda (\(\lambda\)), given a vector of observations. This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where \(\gamma = 1\) and \(\delta = 0\). This function is suitable for maximum likelihood estimation.

Usage

llekw(par, data)

Value

Returns a single double value representing the negative log-likelihood (\(-\ell(\theta|\mathbf{x})\)). Returns Inf

if any parameter values in par 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 3 containing the distribution parameters in the order: alpha (\(\alpha > 0\)), beta (\(\beta > 0\)), 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 Exponentiated Kumaraswamy (EKw) distribution is the GKw distribution (dekw) with \(\gamma=1\) and \(\delta=0\). Its probability density function (PDF) is: $$ f(x | \theta) = \lambda \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} \bigl[1 - (1 - x^\alpha)^\beta \bigr]^{\lambda - 1} $$ for \(0 < x < 1\) and \(\theta = (\alpha, \beta, \lambda)\). The log-likelihood function \(\ell(\theta | \mathbf{x})\) for a sample \(\mathbf{x} = (x_1, \dots, x_n)\) is \(\sum_{i=1}^n \ln f(x_i | \theta)\): $$ \ell(\theta | \mathbf{x}) = n[\ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i)] $$ where:

• \(v_i = 1 - x_i^{\alpha}\)

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

This function computes and returns the negative log-likelihood, \(-\ell(\theta|\mathbf{x})\), suitable for minimization using optimization routines like optim. Numerical stability is maintained similarly to llgkw.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

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

llgkw (parent distribution negative log-likelihood), dekw, pekw, qekw, rekw, grekw (gradient, if available), hsekw (Hessian, if available), optim

Examples

# \donttest{
# Assuming existence of rekw, grekw, hsekw functions for EKw distribution

# Generate sample data from a known EKw distribution
set.seed(123)
true_par_ekw <- c(alpha = 2, beta = 3, lambda = 0.5)
# Use rekw if it exists, otherwise use rgkw with gamma=1, delta=0
if (exists("rekw")) {
  sample_data_ekw <- rekw(100, alpha = true_par_ekw[1], beta = true_par_ekw[2],
                          lambda = true_par_ekw[3])
} else {
  sample_data_ekw <- rgkw(100, alpha = true_par_ekw[1], beta = true_par_ekw[2],
                         gamma = 1, delta = 0, lambda = true_par_ekw[3])
}
hist(sample_data_ekw, breaks = 20, main = "EKw(2, 3, 0.5) Sample")

# --- Maximum Likelihood Estimation using optim ---
# Initial parameter guess
start_par_ekw <- c(1.5, 2.5, 0.8)

# Perform optimization (minimizing negative log-likelihood)
# Use method="L-BFGS-B" for box constraints if needed (all params > 0)
mle_result_ekw <- stats::optim(par = start_par_ekw,
                               fn = llekw, # Use the EKw neg-log-likelihood
                               method = "BFGS", # Or "L-BFGS-B" with lower=1e-6
                               hessian = TRUE,
                               data = sample_data_ekw)

# Check convergence and results
if (mle_result_ekw$convergence == 0) {
  print("Optimization converged successfully.")
  mle_par_ekw <- mle_result_ekw$par
  print("Estimated EKw parameters:")
  print(mle_par_ekw)
  print("True EKw parameters:")
  print(true_par_ekw)
} else {
  warning("Optimization did not converge!")
  print(mle_result_ekw$message)
}

# --- Compare numerical and analytical derivatives (if available) ---
# Requires 'numDeriv' package and analytical functions 'grekw', 'hsekw'
if (mle_result_ekw$convergence == 0 &&
    requireNamespace("numDeriv", quietly = TRUE) &&
    exists("grekw") && exists("hsekw")) {

  cat("\nComparing Derivatives at EKw MLE estimates:\n")

  # Numerical derivatives of llekw
  num_grad_ekw <- numDeriv::grad(func = llekw, x = mle_par_ekw, data = sample_data_ekw)
  num_hess_ekw <- numDeriv::hessian(func = llekw, x = mle_par_ekw, data = sample_data_ekw)

  # Analytical derivatives (assuming they return derivatives of negative LL)
  ana_grad_ekw <- grekw(par = mle_par_ekw, data = sample_data_ekw)
  ana_hess_ekw <- hsekw(par = mle_par_ekw, data = sample_data_ekw)

  # Check differences
  cat("Max absolute difference between gradients:\n")
  print(max(abs(num_grad_ekw - ana_grad_ekw)))
  cat("Max absolute difference between Hessians:\n")
  print(max(abs(num_hess_ekw - ana_hess_ekw)))

} else {
   cat("\nSkipping derivative comparison for EKw.\n")
   cat("Requires convergence, 'numDeriv' package and functions 'grekw', 'hsekw'.\n")
}

# }