llkw {gkwreg} | R Documentation |
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: |
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")
}