${{}\theta}$ accumulates. On the other hand, jackknife, bootstrap, iterated bootstrap, and indirect inference, generally, approximate the bias by simulating samples from the assumed model or an estimator of $G$, like the empirical distribution function. As a result, and depending on how demanding the computation of $\hat{{}\theta}$ is, simulation-based methods are typically more computationally intensive than analytical methods. Also, implicit, simulation-based methods require special care and ad-hoc considerations when approximating the solution of (\[bias\_equation\]), because the simulation-based estimator of ${{}B}_G({{}\theta})$ is not always differentiable with respect to ${{}\theta}$.

The requirement of differentiation of the log-likelihood or surrogate functions for some of the bias-reduction methods in Table \[br\_characteristics\] is also another area where considerable analytical effort has been devoted to (see, for example, @kosmidis+firth:2009 for multivariate generalized nonlinear models, and @grun+kosmidis+zeileis:2012 for Beta regression models). Nevertheless, differentiation is nowadays a task requiring increasingly less analytical effort because of the availability of comprehensive automatic differentiation routines [@griewank+walther:2008] in popular computing environments; such routines can be found, for example, in the FowardDiff Julia package [@revels+lubin+papamarkou:2016], and the CppAD package for C++ [@bell:2019] that enabled the development of software like the TMB package [@kristensen+nielsen+berg+etal:2016] for R [@rproject] which is a generic framework for fitting and 