Empirical likelihood estimating equations

Usage

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

Details

Returns a function that evaluates the stacked EL system for $\theta = (\beta, z, \lambda_x)$ with $z = \operatorname{logit}(W)$. Blocks correspond to: (i) missingness (response) model score equations in $\beta$, (ii) the response-rate equation in $W$, and (iii) auxiliary moment constraints in $\lambda_x$. When no auxiliaries are present the last block is omitted. The system matches Qin, Leung, and Shao (2002, Eqs. 7-10) with empirical masses $m_i = d_i/D_i(\theta)$, $D_i$ as in the paper. We cap $\eta$, clip $w_i$ in ratios, and guard $D_i$ away from zero to ensure numerical stability; these safeguards are applied consistently in equations, Jacobian, and post-solution weights.

Guarding policy (must remain consistent across equations/Jacobian/post):

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.

The score with respect to the linear predictor uses the Bernoulli form $s_{\eta,i}(\beta) = \partial \log w_i / \partial \eta_i = \mu.\eta(\eta_i)/w_i$, which is valid for both logit and probit links when $w_i$ is clipped.

References

Qin, J., Leung, D., and Shao, J. (2002). Estimation with survey data under nonignorable nonresponse or informative sampling. Journal of the American Statistical Association, 97(457), 193-200.