Analytical Jacobian for empirical likelihood
el_build_jacobian(
family,
missingness_model_matrix,
auxiliary_matrix,
respondent_weights,
N_pop,
n_resp_weighted,
mu_x_scaled
)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.
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.