adjustments to the estimating functions in Section \[sec:theory\]. Such an assumption covers many well-used estimating functions, like the ones arising in quasi-likelihood estimation, estimation using generalized estimating equations, and maximum likelihood and maximum composite likelihood estimation for a wide range of models. The smoothness assumption does not cover, though, settings where the estimating function or one of its first few derivatives are non-differentiable at particular points in the parameter space. Examples of this kind are the estimating functions for quantile regression and robust regression with Huber loss; see @koenker:2005 and @huber+ronchetti:2009 for textbook-length expositions of topics in quantile and robust regression, respectively.

Assumptions \[derivative\_orders\] and \[moment\_orders\] ensure the existence of the expectations, under the underlying process $G$, of products of estimating functions and their derivatives, and that $\sqrt{n}$-asymptotic arguments are valid. Assumption \[invertibility\] is a technical assumption to ensure that the expectation of the jacobian of the estimating function is invertible, when inverting the stochastic Taylor expansions in Section \[sec:theory\], and is typically assumed for estimation using maximum likelihood and estimating equations [see, for example, @boos+stefanski:2013 Section 7.7].

Adjusted estimating equations for bias reduction {#sec:theory}
================================================

Asymptotic bias {#asymptotic_bias}
---------------

Under the assumptions of Section \[assumptions\] it can be shown 