\right\} \psi_t^i({{}\theta}) \, .$$

The above expressions can directly be used in settings that involve observations of $k$ independent random vectors with dependent components. Examples of such settings are the generalized estimating equations in @liang+zeger:1986 for estimating marginal regression parameters for correlated responses, and the composite likelihood approach in @varin+czado:2010 for estimating multivariate probit models with random effects to account for serial dependence in longitudinal settings with binary and ordinal outcomes. Section \[autoregressive\_probit\] considers a case study using the latter models.

Reduced-bias $M$-estimation for partially-specified models
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Under the assumptions of Section \[assumptions\] and the assumptions about $G$ required for the specification of ${{}e}({{}\theta})$ and ${{}d}_r({{}\theta})$ in Section \[independence\], the solution of the adjusted estimating equations $\sum_{i=1}^k{{}\psi}^i({{}\theta}) + {{}A}({{}\theta}) =
0_p$, with ${{}A}({{}\theta})$ as in (\[general\_empirical\_adjustment\]), will result in RB$M$-estimators that have $O(n^{-3/2})$ bias. This is a major advantage over past bias-reduction methods [see, e.g. @cordeiro+mccullagh:1991; @firth:1993] whose applicability is limited to cases where $\sum_{i=1}^k{{}\psi}^i({{}\theta})$ is the gradient of the log-likelihood function of a correctly-specified model.

The example below demonstrates the benefits of using the empirical bias-reducing adjustment in the estimation of a ratio of means from realizations of independent pairs of random variables without any further assumptions on the 