llekw: Negative Log-Likelihood for the Exponentiated Kumaraswamy...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

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)

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).

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.

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).

Author(s)

Lopes, J. E.

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

# 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")
}

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