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Fit univariate beta distribution via weighted MLE

fit_beta_mle_single.Rd

Fits a Beta(alpha, beta) distribution to univariate data using weighted maximum likelihood estimation. Initializes parameters via weighted moment matching, then refines estimates using Newton-Raphson optimization.

Usage

fit_beta_mle_single(x, w, tol = 1e-06, max_iter = 100, clip_eps = NULL)

Arguments

x

numeric vector of observations in the interval (0, 1)

w

numeric vector of normalized weights that sum to 1. Must have the same length as x

tol

numeric convergence tolerance for gradient norm (default: 1e-6)

max_iter

integer maximum number of Newton-Raphson iterations (default: 100)

clip_eps

numeric clipping epsilon for numerical stability. Values outside [clip_eps, 1 - clip_eps] are clipped. Default: same as tol

Value

A list with components:

alpha

fitted shape parameter (> 0)

beta

fitted shape parameter (> 0)

converged

logical indicating whether optimization converged

iterations

number of Newton-Raphson iterations performed

Details

The function optimizes the weighted log-likelihood: $$\sum_i w_i \log f(x_i; \alpha, \beta)$$ where \(f(x; \alpha, \beta)\) is the beta density.

Algorithm:

  1. Clip x to [clip_eps, 1 - clip_eps] for numerical stability

  2. Initialize via weighted method of moments

  3. Iteratively update parameters using Newton-Raphson with backtracking line search

  4. Convergence declared when gradient norm < tol

Numerical considerations:

  • Returns alpha = beta = 100 for degenerate cases (variance < 1e-10)

  • Uses backtracking line search to maintain positivity of parameters

  • Warns if Hessian becomes singular (determinant < 1e-10)

  • Warns if maximum iterations reached without convergence

Examples

if (FALSE) { # \dontrun{
# Unweighted fit
x <- rbeta(100, 2, 5)
w <- rep(1/100, 100)
fit <- fit_beta_mle_single(x, w)

# Weighted fit
x <- rbeta(100, 3, 3)
w <- runif(100)
w <- w / sum(w)
fit <- fit_beta_mle_single(x, w, tol = 1e-8, max_iter = 200)
} # }