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

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

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)

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).

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.

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

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

# 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")
}

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