,$$ in (\[adjusted\_estimating\_equations\]), will result in estimators with bias $\expect_G{(\tilde{\theta}^{r} - \bar{\theta}^{r})} = O(n^{-3/2})$, which is smaller, asymptotically, than the bias of $\hat{{}\theta}$ in (\[bias\]). Hence, expression (\[adjustment\]) defines a family of bias-reducing adjustments to estimating functions.

A clear candidate for ${{}A}$ has as $r$th component the first term in the right hand side of (\[adjustment\]). If we assume that the model $F_i({{}y}_i | {{}x}_i, {{}\theta})$ fully specifies $G$ and the estimation method is maximum likelihood, then the Bartlett relations $\mu_{cd} + \nu_{c, d} = 0$ hold [see, for example, @pace+salvan:1997 Section 9.2], and the adjustment (\[adjustment\]) becomes ${\mu^{ab}_{}} \left( 2 {\nu^{}_{ra,b}} + {\mu^{}_{rab}}
\right)/2$. The latter expression coincides with the bias-reducing score adjustment derived in @firth:1993.

For the more general setting of Section \[modelling\_setting\], however, the underlying distribution $G$ is at most only partially specified through $F_i({{}y}_i | {{}x}_i, {{}\theta})$, and the expectations involved in the right-hand side of (\[adjustment\]) cannot be computed. It turns out that, there is a family of empirical adjustments that can always be implemented and delivers bias reduction of $M$-estimators in that more general framework.

Empirical bias-reducing adjustments {#empirical_adjustments}
===================================

Derivation
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Consider data-dependent quantities ${l^{}_{r|s}}({{}\theta})$ and ${l^{}_{rs|t}}({{}\theta})$ such that $\expect_G({l^{}_{r|s}}) = {\nu^{}_{r, s}}$ 