Analytical Jacobian for empirical likelihood

Usage

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

Details

Builds the block Jacobian $A = \partial F/\partial \theta$ for the EL system with $\theta = (\beta, z, \lambda_x)$ and $z = \operatorname{logit}(W)$. Blocks follow Qin, Leung, and Shao (2002, Eqs. 7-10). The derivative with respect to the linear predictor for the missingness (response) model uses the Bernoulli score form $\partial/\partial\eta\, \log w(\eta) = \mu.\eta(\eta)/w(\eta)$ with link-inverse clipping. Denominator guards are applied consistently when forming terms depending on $D_i(\theta)$.

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()). Terms that depend on d(1/Di)/d(.) are multiplied by active = 1(Di_raw > floor) to match the clamped equations.

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.