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Empirical likelihood estimating equations for SRS

Source: R/engines_el_impl_equations.R
el_build_equation_system.Rd

Returns a function that evaluates the stacked EL system for \(\theta = (\beta, z, \lambda_x)\) with \(z = \operatorname{logit}(W)\). Blocks correspond to:

  1. missingness model score equations in \(\beta\),

  2. the response-rate equation in \(W\),

  3. auxiliary moment constraints in \(\lambda_x\).

Usage

el_build_equation_system(
  family,
  missingness_model_matrix,
  auxiliary_matrix,
  respondent_weights,
  N_pop,
  n_resp_weighted,
  mu_x_scaled
)

Details

When no auxiliaries are present the last block is omitted. The system matches QLS equations 7-10. We cap \(\eta\), clip \(w_i\) in ratios, and guard \(D_i\) away from zero to ensure numerical stability.

Guarding policy:

  • Cap \(\eta\): eta <- pmax(pmin(eta, get_eta_cap()), -get_eta_cap()).

  • Compute w <- family$linkinv(eta) and clip to [1e-12, 1 - 1e-12] when used in ratios.

  • Denominator floor: Di <- pmax(Di_raw, nmar_get_el_denom_floor()). In the Jacobian, terms that depend on d(1/Di)/d(.) are multiplied by active = 1(Di_raw > floor) to match the clamped equations.