is the same as that of the original $M$-estimator, and introduces Wald-type and generalized score approximate pivots that can be used for the construction of inferences. Section \[sec:penalties\] introduces and discusses bias-reducing penalized objectives and the links to model selection. A quasi Newton-Raphson iteration that has the BR$M$-estimates as its fixed point is introduced in Section \[sec:implementation\], along with discussion on its ingredients and on general implementations using automatic differentiation. The finite-sample properties of RB$M$-estimation are illustrated in well-used, important modelling settings of increasing complexity including, the estimation of the ratio of two means with minimal distributional assumptions (Section \[empirical\_adjustments\]), generalized linear models (Section \[sec:glms\]) and quasi likelihoods (Section \[sec:quasi\]), and composite likelihood methods for the estimation of gaussian max-stable processes (Section \[sec:maxstable\]), and for the estimation of multivariate probit models with correlated random effects for longitudinal ordinal responses (Section \[autoregressive\_probit\]). Section \[sec:discussion\] provides a discussion on the developments and possible extensions of the developments in this work, and introduces two alternative RB$M$-estimators.

Modelling setting and assumptions {#modelling_setting}
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Estimating functions {#EF}
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Suppose that we observe the values ${{}y}_1, \ldots, {{}y}_k$ of a sequence of random vectors ${{}Y}_1, \ldots, {{}Y}_k$ with ${{}y}_i = (y_{i1}, \ldots, y_{ic_i})^\top \in \mathcal{Y} \subset
