inference and model selection with reduced-bias $M$-estimates. The properties of reduced-bias $M$-estimation are illustrated in well-used, important modelling settings of varying complexity.\
    [Keywords: *automatic differentiation*; *composite likelihood*; *model selection*; *penalized likelihood*; *quasi likelihood*; *quasi Newton-Raphson*]{}
author:
- |
    Ioannis Kosmidis\
    `ioannis.kosmidis@warwick.ac.uk`\
    Department of Statistics, University of Warwick\
    Gibbet Hill Road, Coventry, CV4 7AL, UK\
    and\
    The Alan Turing Institute\
    British Library, 96 Euston Road, London, NW1 2DB, UK\
    and\
    Nicola Lunardon\
    `nicola.lunardon@unimib.it`\
    Department of Economics, Management and Statistics, University of Milano-Bicocca\
    Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
bibliography:
- 'robustbr.bib'
title: 'Empirical bias-reducing adjustments to estimating functions'
---

=1

Introduction
============

Reduction of estimation bias in statistical modelling is a task that has attracted immense research activity since the early days of the statistical literature. This ongoing activity resulted in an abundance of general bias-reduction methods of wide applicability. As is noted in @kosmidis:2014, the majority of those methods start from an estimator $\hat{{}\theta}$ and, directly or indirectly, attempt to produce an estimator $\tilde{{}\theta}$ of an unknown parameter ${{}\theta}$, which approximates the solution of the equation $$\label{bias_equation}
  \hat{{}\theta} - \tilde{{}\theta} = {{}B}_G(\bar{{}\theta}) \, ,$$ with respect to $\tilde{{}\theta}$. In the above equation, $G$ is the typically unknown, joint 