llbkw: Negative Log-Likelihood for Beta-Kumaraswamy (BKw)...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Negative Log-Likelihood for Beta-Kumaraswamy (BKw) Distribution

Description

Computes the negative log-likelihood function for the Beta-Kumaraswamy (BKw) distribution with parameters alpha (\alpha), beta (\beta), gamma (\gamma), and delta (\delta), given a vector of observations. This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where \lambda = 1. This function is typically used for maximum likelihood estimation via numerical optimization.

Usage

llbkw(par, data)

Arguments

par	A numeric vector of length 4 containing the distribution parameters in the order: alpha (\alpha > 0), beta (\beta > 0), gamma (\gamma > 0), delta (\delta \ge 0).
data	A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Details

The Beta-Kumaraswamy (BKw) distribution is the GKw distribution (dgkw) with \lambda=1. Its probability density function (PDF) is:

f(x | \theta) = \frac{\alpha \beta}{B(\gamma, \delta+1)} x^{\alpha - 1} \bigl(1 - x^\alpha\bigr)^{\beta(\delta+1) - 1} \bigl[1 - \bigl(1 - x^\alpha\bigr)^\beta\bigr]^{\gamma - 1}

for 0 < x < 1, \theta = (\alpha, \beta, \gamma, \delta), and B(a,b) is the Beta function (beta). 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(\alpha) + \ln(\beta) - \ln B(\gamma, \delta+1)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta(\delta+1)-1)\ln(v_i) + (\gamma-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

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), dbkw, pbkw, qbkw, rbkw, grbkw (gradient, if available), hsbkw (Hessian, if available), optim, lbeta

Examples

# Generate sample data from a known BKw distribution
set.seed(2203)
true_par_bkw <- c(alpha = 2.0, beta = 1.5, gamma = 1.5, delta = 0.5)
sample_data_bkw <- rbkw(1000, alpha = true_par_bkw[1], beta = true_par_bkw[2],
                         gamma = true_par_bkw[3], delta = true_par_bkw[4])
hist(sample_data_bkw, breaks = 20, main = "BKw(2, 1.5, 1.5, 0.5) Sample")

# --- Maximum Likelihood Estimation using optim ---
# Initial parameter guess
start_par_bkw <- c(1.8, 1.2, 1.1, 0.3)

# Perform optimization (minimizing negative log-likelihood)
mle_result_bkw <- stats::optim(par = start_par_bkw,
                               fn = llbkw, # Use the BKw neg-log-likelihood
                               method = "BFGS", # Needs parameters > 0, consider L-BFGS-B
                               hessian = TRUE,
                               data = sample_data_bkw)

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

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

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

  # Numerical derivatives of llbkw
  num_grad_bkw <- numDeriv::grad(func = llbkw, x = mle_par_bkw, data = sample_data_bkw)
  num_hess_bkw <- numDeriv::hessian(func = llbkw, x = mle_par_bkw, data = sample_data_bkw)

  # Analytical derivatives (assuming they return derivatives of negative LL)
  ana_grad_bkw <- grbkw(par = mle_par_bkw, data = sample_data_bkw)
  ana_hess_bkw <- hsbkw(par = mle_par_bkw, data = sample_data_bkw)

  # Check differences
  cat("Max absolute difference between gradients:\n")
  print(max(abs(num_grad_bkw - ana_grad_bkw)))
  cat("Max absolute difference between Hessians:\n")
  print(max(abs(num_hess_bkw - ana_hess_bkw)))

} else {
   cat("\nSkipping derivative comparison for BKw.\n")
   cat("Requires convergence, 'numDeriv' package and functions 'grbkw', 'hsbkw'.\n")
}

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