gkwreg (version 1.0.7)

llkw: Negative Log-Likelihood of the Kumaraswamy (Kw) Distribution

Description

Computes the negative log-likelihood function for the two-parameter Kumaraswamy (Kw) distribution with parameters alpha (\(\alpha\)) and beta (\(\beta\)), given a vector of observations. This function is suitable for maximum likelihood estimation.

Usage

llkw(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 2 containing the distribution parameters in the order: alpha (\(\alpha > 0\)), beta (\(\beta > 0\)).

data

A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Author

Lopes, J. E.

Details

The Kumaraswamy (Kw) distribution's probability density function (PDF) is (see dkw): $$ f(x | \theta) = \alpha \beta x^{\alpha-1} (1 - x^\alpha)^{\beta-1} $$ for \(0 < x < 1\) and \(\theta = (\alpha, \beta)\). The log-likelihood function \(\ell(\theta | \mathbf{x})\) for a sample \(\mathbf{x} = (x_1, \dots, x_n)\) is \(\sum_{i=1}^n \ln f(x_i | \theta)\): $$ \ell(\theta | \mathbf{x}) = n[\ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i)] $$ where \(v_i = 1 - x_i^{\alpha}\). This function computes and returns the negative log-likelihood, \(-\ell(\theta|\mathbf{x})\), suitable for minimization using optimization routines like optim. It is equivalent to the negative log-likelihood of the GKw distribution (llgkw) evaluated at \(\gamma=1, \delta=0, \lambda=1\).

References

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

See Also

llgkw (parent distribution negative log-likelihood), dkw, pkw, qkw, rkw, grkw (gradient, if available), hskw (Hessian, if available), optim

Examples

# \donttest{
# Assuming existence of rkw, grkw, hskw functions for Kw distribution

# Generate sample data from a known Kw distribution
set.seed(123)
true_par_kw <- c(alpha = 2, beta = 3)
sample_data_kw <- rkw(100, alpha = true_par_kw[1], beta = true_par_kw[2])
hist(sample_data_kw, breaks = 20, main = "Kw(2, 3) Sample")

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

# Perform optimization (minimizing negative log-likelihood)
# Use method="L-BFGS-B" for box constraints (params > 0)
mle_result_kw <- stats::optim(par = start_par_kw,
                              fn = llkw, # Use the Kw neg-log-likelihood
                              method = "L-BFGS-B",
                              lower = c(1e-6, 1e-6), # Lower bounds > 0
                              hessian = TRUE,
                              data = sample_data_kw)

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

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

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

  # Numerical derivatives of llkw
  num_grad_kw <- numDeriv::grad(func = llkw, x = mle_par_kw, data = sample_data_kw)
  num_hess_kw <- numDeriv::hessian(func = llkw, x = mle_par_kw, data = sample_data_kw)

  # Analytical derivatives (assuming they return derivatives of negative LL)
  ana_grad_kw <- grkw(par = mle_par_kw, data = sample_data_kw)
  ana_hess_kw <- hskw(par = mle_par_kw, data = sample_data_kw)

  # Check differences
  cat("Max absolute difference between gradients:\n")
  print(max(abs(num_grad_kw - ana_grad_kw)))
  cat("Max absolute difference between Hessians:\n")
  print(max(abs(num_hess_kw - ana_hess_kw)))

} else {
   cat("\nSkipping derivative comparison for Kw.\n")
   cat("Requires convergence, 'numDeriv' package and functions 'grkw', 'hskw'.\n")
}

# }