${{}b}({{}\theta})$ and, hence, compute $\hat{{}\theta} - {{}b}(\hat{{}\theta})$ in light of data, because the expectations involved in the joint null moments are with respect to the modelling assumption. This is also the basis of more refined bias reduction methods, like the adjusted score function approach that has been derived in @firth:1993 and explored further in @kosmidis+firth:2009.

In the more general setting of Section \[modelling\_setting\], where the model can be only partially specified, naive evaluation of ${{}b}({{}\theta})$ using $F_i({{}y}_i | {{}x}_i, {{}\theta})$ not only does not lead to reduction of bias, in general, but it can also inflate the bias; see, for example, @lunardon+scharfstein:2017 [Section 2.1] who illustrate the impact of using an incorrect bias function in the estimation of log-odds in longitudinal settings. Furthermore, even in the case that the researcher is comfortable to assume that $F_i({{}y}_i | {{}x}_i, {{}\theta})$ is correctly specified, the applicability of standard bias-reduction methods is hampered whenever $F_i({{}y}_i | {{}x}_i, {{}\theta})$ is impossible or impractical to compute in closed form. In such cases, estimation and inference is based on pseudo likelihoods or estimating functions; an example of this kind is within the framework of max-stable processes for which the evaluation of the joint density becomes quickly 