grbeta: Gradient of the Negative Log-Likelihood for the Beta...
| grbeta | R Documentation |
Gradient of the Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization)
Description
Computes the gradient vector (vector of first partial derivatives) of the
negative log-likelihood function for the standard Beta distribution, using
a parameterization common in generalized distribution families. The
distribution is parameterized by gamma (\gamma) and delta
(\delta), corresponding to the standard Beta distribution with shape
parameters shape1 = gamma and shape2 = delta + 1.
The gradient is useful for optimization algorithms.
Usage
grbeta(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
This function calculates the gradient of the negative log-likelihood for a
Beta distribution with parameters shape1 = gamma (\gamma) and
shape2 = delta + 1 (\delta+1). The components of the gradient
vector (-\nabla \ell(\theta | \mathbf{x})) are:
-\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] -
\sum_{i=1}^{n}\ln(x_i)
-\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] -
\sum_{i=1}^{n}\ln(1-x_i)
where \psi(\cdot) is the digamma function (digamma).
These formulas represent the derivatives of -\ell(\theta), consistent with
minimizing the negative log-likelihood. They correspond to the relevant components
of the general GKw gradient (grgkw) evaluated at \alpha=1, \beta=1, \lambda=1.
Note the parameterization: the standard Beta shape parameters are \gamma and \delta+1.
Value
Returns a numeric vector of length 2 containing the partial derivatives
of the negative log-likelihood function -\ell(\theta | \mathbf{x}) with
respect to each parameter: (-\partial \ell/\partial \gamma, -\partial \ell/\partial \delta).
Returns a vector of NaN if any parameter values 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
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd ed.). Wiley.
Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation,
(Note: Specific gradient formulas might be derived or sourced from additional references).
See Also
grgkw, grmc (related gradients),
llbeta (negative log-likelihood function),
hsbeta (Hessian, if available),
dbeta_, pbeta_, qbeta_, rbeta_,
optim,
grad (for numerical gradient comparison),
digamma.
Examples
# Assuming existence of rbeta_, llbeta, grbeta, hsbeta functions
# Generate sample data from a Beta(2, 4) distribution
# (gamma=2, delta=3 in this parameterization)
set.seed(123)
true_par_beta <- c(gamma = 2, delta = 3)
sample_data_beta <- rbeta_(100, gamma = true_par_beta[1], delta = true_par_beta[2])
hist(sample_data_beta, breaks = 20, main = "Beta(2, 4) Sample")
# --- Find MLE estimates ---
start_par_beta <- c(1.5, 2.5)
mle_result_beta <- stats::optim(par = start_par_beta,
fn = llbeta,
gr = grbeta, # Use analytical gradient
method = "L-BFGS-B",
lower = c(1e-6, 1e-6), # Bounds: gamma>0, delta>=0
hessian = TRUE,
data = sample_data_beta)
# --- Compare analytical gradient to numerical gradient ---
if (mle_result_beta$convergence == 0 &&
requireNamespace("numDeriv", quietly = TRUE)) {
mle_par_beta <- mle_result_beta$par
cat("\nComparing Gradients for Beta at MLE estimates:\n")
# Numerical gradient of llbeta
num_grad_beta <- numDeriv::grad(func = llbeta, x = mle_par_beta, data = sample_data_beta)
# Analytical gradient from grbeta
ana_grad_beta <- grbeta(par = mle_par_beta, data = sample_data_beta)
cat("Numerical Gradient (Beta):\n")
print(num_grad_beta)
cat("Analytical Gradient (Beta):\n")
print(ana_grad_beta)
# Check differences
cat("Max absolute difference between Beta gradients:\n")
print(max(abs(num_grad_beta - ana_grad_beta)))
} else {
cat("\nSkipping Beta gradient comparison.\n")
}
# Example with Hessian comparison (if hsbeta exists)
if (mle_result_beta$convergence == 0 &&
requireNamespace("numDeriv", quietly = TRUE) && exists("hsbeta")) {
num_hess_beta <- numDeriv::hessian(func = llbeta, x = mle_par_beta, data = sample_data_beta)
ana_hess_beta <- hsbeta(par = mle_par_beta, data = sample_data_beta)
cat("\nMax absolute difference between Beta Hessians:\n")
print(max(abs(num_hess_beta - ana_hess_beta)))
}
