and $\expect_G({l^{}_{rs|t}}) = {\nu^{}_{rs, t}}$, with ${H^{}_{r | s}} = {l^{}_{r | s}} - {\nu^{}_{r, s}} = O_p(n^{1/2})$ and ${H^{}_{rs | t}} = {l^{}_{rs | t}} - {\nu^{}_{rs, t}} =
O_p(n^{1/2})$. Then, a simple calculation under the assumptions of Section \[assumptions\] gives that equation (\[adjustment\]) is satisfied by an empirical adjustment of the form $$\label{general_empirical_adjustment}
  A_r({{}\theta}) = \frac{1}{2} {l^{ab}_{}}({{}\theta}) \left\{ 2 {l^{}_{ra|b}}({{}\theta}) - {l^{cd}_{}}({{}\theta}) {l^{}_{b|d}}({{}\theta}) {l^{}_{rac}}({{}\theta})\right\} \, ,$$ where ${l^{st}_{}}({{}\theta})$ is the matrix inverse of ${l^{}_{ts}}({{}\theta})$. The matrix form of expression (\[general\_empirical\_adjustment\]) sets the $r$th element of the vector of empirical bias-reducing adjustments to $$\label{general_empirical_adjustment_matrix}
  A_r({{}\theta}) = - \trace\left\{{{}j}({{}\theta})^{-1} {{}d}_r({{}\theta}) \right\} -
  \frac{1}{2}\trace\left[ {{}j}({{}\theta})^{-1} {{}e}({{}\theta}) \left\{{{}j}({{}\theta})^{-1}\right\}^\top {{}u}_r({{}\theta}) \right] \, ,$$ where ${{}u}_r({{}\theta})=\sum_{i=1}^k \nabla \nabla^\top \psi_r^i({{}\theta})$, and ${{}j}({{}\theta})$ is the matrix with $s$th row $-\sum_{i=1}^k \nabla \psi_s^i({{}\theta})$, assumed to be invertible but not necessarily symmetric. The matrix ${{}j}({{}\theta})$ coincides with the negative hessian matrix $\nabla \nabla^\top l({{}\theta})$ when estimation is through the maximisation of an objective function like (\[objective\_function\]), and is the observed information matrix when estimation is through maximum likelihood. The $p \times p$ matrices ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$ correspond to the quantities ${l^{}_{r|s}}({{}\theta})$ and ${l^{}_{rs|t}}({{}\theta})$, respectively, and need to be exactly defined in order to use adjustment (\[general\_empirical\_adjustment\]) in estimation 