{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T22:21:29Z","timestamp":1789683689087,"version":"4.0.1"},"reference-count":69,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T00:00:00Z","timestamp":1694736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,2,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We develop a novel, general framework for reduced-bias M-estimation from asymptotically unbiased estimating functions. The framework relies on an empirical approximation of the bias by a function of derivatives of estimating function contributions. Reduced-bias M-estimation operates either implicitly, solving empirically adjusted estimating equations, or explicitly, subtracting the estimated bias from the original M-estimates, and applies to partially or fully specified models with likelihoods or surrogate objectives. Automatic differentiation can abstract away the algebra required to implement reduced-bias M-estimation. As a result, the bias-reduction methods, we introduce have broader applicability, straightforward implementation, and less algebraic or computational effort than other established bias-reduction methods that require resampling or expectations of products of log-likelihood derivatives. If M-estimation is by maximising an objective, then there always exists a bias-reducing penalised objective. That penalised objective relates to information criteria for model selection and can be enhanced with plug-in penalties to deliver reduced-bias M-estimates with extra properties, like finiteness for categorical data models. Inferential procedures and model selection procedures for M-estimators apply unaltered with the reduced-bias M-estimates. We demonstrate and assess the properties of reduced-bias M-estimation in well-used, prominent modelling settings of varying complexity.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad083","type":"journal-article","created":{"date-parts":[[2023,9,16]],"date-time":"2023-09-16T00:44:25Z","timestamp":1694825065000},"page":"62-89","source":"Crossref","is-referenced-by-count":7,"title":["Empirical bias-reducing adjustments to estimating functions"],"prefix":"10.1093","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1556-0302","authenticated-orcid":false,"given":"Ioannis","family":"Kosmidis","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Warwick , Gibbet Hill Road, Coventry CV4 7AL , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9145-7185","authenticated-orcid":false,"given":"Nicola","family":"Lunardon","sequence":"additional","affiliation":[{"name":"Department of Environmental Sciences, Informatics and Statistics, Ca\u2019 Foscari University of Venice , Via Torino 150, Venezia Mestre 30170 , Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,9,16]]},"reference":[{"issue":"6","key":"2024020822013990800_qkad083-B1","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"1","key":"2024020822013990800_qkad083-B2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/71.1.1","article-title":"On the existence of maximum likelihood estimates in logistic regression models","volume":"71","author":"Albert","year":"1984","journal-title":"Biometrika"},{"key":"2024020822013990800_qkad083-B3","author":"Bell","year":"2021"},{"issue":"2","key":"2024020822013990800_qkad083-B4","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1111\/j.2517-6161.1974.tb00999.x","article-title":"Spatial interaction and the statistical analysis of lattice systems","volume":"36","author":"Besag","year":"1974","journal-title":"Journal of the Royal Statistical Society. Series B (Methodological)"},{"issue":"3","key":"2024020822013990800_qkad083-B5","first-page":"179","article-title":"Statistical analysis of non-lattice data","volume":"24","author":"Besag","year":"1975","journal-title":"Journal of the Royal Statistical Society. Series D (The Statistician)"},{"issue":"1","key":"2024020822013990800_qkad083-B6","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1111\/j.2517-6161.1972.tb00889.x","article-title":"Nearest-neighbour systems and the auto-logistic model for binary data","volume":"34","author":"Besag","year":"1972","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"issue":"4","key":"2024020822013990800_qkad083-B7","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1080\/00031305.1992.10475921","article-title":"On generalized score tests","volume":"46","author":"Boos","year":"1992","journal-title":"The American Statistician"},{"key":"2024020822013990800_qkad083-B8","author":"Canty","year":"2022"},{"issue":"3","key":"2024020822013990800_qkad083-B9","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1198\/jabes.2009.07032","article-title":"Autologistic models with interpretable parameters","volume":"14","author":"Caragea","year":"2009","journal-title":"Journal of Agricultural, Biological, and Environmental Statistics"},{"issue":"3","key":"2024020822013990800_qkad083-B10","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1214\/aos\/1176350057","article-title":"The use of subseries values for estimating the variance of a general statistic from a stationary sequence","volume":"14","author":"Carlstein","year":"1986","journal-title":"The Annals of Statistics"},{"key":"2024020822013990800_qkad083-B11","volume-title":"Model selection and model averaging","author":"Claeskens","year":"2008"},{"issue":"3","key":"2024020822013990800_qkad083-B12","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1111\/j.2517-6161.1991.tb01852.x","article-title":"Bias correction in generalized linear models","volume":"53","author":"Cordeiro","year":"1991","journal-title":"Journal of the Royal Statistical Society. Series B (Methodological)"},{"issue":"2138","key":"2024020822013990800_qkad083-B13","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1098\/rspa.2011.0412","article-title":"Geostatistics of extremes","volume":"468","author":"Davison","year":"2012","journal-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences"},{"key":"2024020822013990800_qkad083-B14","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511802843","volume-title":"Bootstrap methods and their application","author":"Davison","year":"1997"},{"issue":"3","key":"2024020822013990800_qkad083-B15","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1093\/biomet\/46.3-4.477","article-title":"A note on the application of Quenouille\u2019s method of bias reduction to the estimation of ratios","volume":"46","author":"Durbin","year":"1959","journal-title":"Biometrika"},{"issue":"6","key":"2024020822013990800_qkad083-B16","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1214\/aos\/1176343282","article-title":"Defining the curvature of a statistical problem (with applications to second order efficiency)","volume":"3","author":"Efron","year":"1975","journal-title":"The Annals of Statistics"},{"key":"2024020822013990800_qkad083-B17","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970319","volume-title":"The jackknife, the bootstrap and other resampling plans","author":"Efron","year":"1982"},{"key":"2024020822013990800_qkad083-B18","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-4541-9","volume-title":"An introduction to the bootstrap","author":"Efron","year":"1993"},{"issue":"1","key":"2024020822013990800_qkad083-B19","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/biomet\/80.1.27","article-title":"Bias reduction of maximum likelihood estimates","volume":"80","author":"Firth","year":"1993","journal-title":"Biometrika"},{"issue":"2","key":"2024020822013990800_qkad083-B20","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1093\/biomet\/asr020","article-title":"On the likelihood function of gaussian max-stable processes","volume":"98","author":"Genton","year":"2011","journal-title":"Biometrika"},{"issue":"S1","key":"2024020822013990800_qkad083-B21","doi-asserted-by":"crossref","first-page":"S85","DOI":"10.1002\/jae.3950080507","article-title":"Indirect inference","volume":"8","author":"Gourieroux","year":"1993","journal-title":"Journal of Applied Econometrics"},{"key":"2024020822013990800_qkad083-B22","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898717761","volume-title":"Evaluating derivatives: Principles and techniques of algorithmic differentiation","author":"Griewank","year":"2008","edition":"2nd ed."},{"issue":"11","key":"2024020822013990800_qkad083-B23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v048.i11","article-title":"Extended beta regression in R: Shaken, stirred, mixed, and partitioned","volume":"48","author":"Gr\u00fcn","year":"2012","journal-title":"Journal of Statistical Software"},{"issue":"525","key":"2024020822013990800_qkad083-B24","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1080\/01621459.2017.1380031","article-title":"Simulation-based bias correction methods for complex models","volume":"114","author":"Guerrier","year":"2019","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2024020822013990800_qkad083-B25","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1093\/biomet\/75.4.661","article-title":"On bootstrap resampling and iteration","volume":"75","author":"Hall","year":"1988","journal-title":"Biometrika"},{"issue":"449","key":"2024020822013990800_qkad083-B26","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/01621459.2000.10473914","article-title":"Window subsampling of estimating functions with application to regression models","volume":"95","author":"Heagerty","year":"2000","journal-title":"Journal of the American Statistical Association"},{"issue":"7","key":"2024020822013990800_qkad083-B27","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1002\/env.1102","article-title":"Autologistic models for binary data on a lattice","volume":"22","author":"Hughes","year":"2011","journal-title":"Environmetrics"},{"issue":"2","key":"2024020822013990800_qkad083-B28","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1093\/biomet\/ass089","article-title":"Composite likelihood estimation for the Brown\u2013Resnick process","volume":"100","author":"Huser","year":"2013","journal-title":"Biometrika"},{"issue":"4","key":"2024020822013990800_qkad083-B29","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1093\/biomet\/asx046","article-title":"Median bias reduction of maximum likelihood estimates","volume":"104","author":"Kenne Pagui","year":"2017","journal-title":"Biometrika"},{"key":"2024020822013990800_qkad083-B30","author":"Konis","year":"2007"},{"issue":"3","key":"2024020822013990800_qkad083-B31","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1002\/wics.1296","article-title":"Bias in parametric estimation: Reduction and useful side-effects","volume":"6","author":"Kosmidis","year":"2014","journal-title":"Wiley Interdisciplinary Reviews: Computational Statistics"},{"key":"2024020822013990800_qkad083-B32","author":"Kosmidis","year":"2023"},{"issue":"4","key":"2024020822013990800_qkad083-B33","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1093\/biomet\/asp055","article-title":"Bias reduction in exponential family nonlinear models","volume":"96","author":"Kosmidis","year":"2009","journal-title":"Biometrika"},{"issue":"1","key":"2024020822013990800_qkad083-B34","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1093\/biomet\/asaa052","article-title":"Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models","volume":"108","author":"Kosmidis","year":"2021","journal-title":"Biometrika"},{"issue":"1","key":"2024020822013990800_qkad083-B35","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s11222-019-09860-6","article-title":"Mean and median bias reduction in generalized linear models","volume":"30","author":"Kosmidis","year":"2020","journal-title":"Statistics and Computing"},{"key":"2024020822013990800_qkad083-B36","author":"Kosmidis","year":"2022"},{"key":"2024020822013990800_qkad083-B37","author":"Kosmidis","year":"2022"},{"issue":"5","key":"2024020822013990800_qkad083-B38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v070.i05","article-title":"TMB: Automatic differentiation and Laplace approximation","volume":"70","author":"Kristensen","year":"2016","journal-title":"Journal of Statistical Software"},{"issue":"2","key":"2024020822013990800_qkad083-B39","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1111\/j.2517-6161.1995.tb02035.x","article-title":"Asymptotically unbiased estimation in generalized linear models with random effects","volume":"57","author":"Kuk","year":"1995","journal-title":"Journal of the Royal Statistical Society. Series B (Methodological)"},{"issue":"3","key":"2024020822013990800_qkad083-B40","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1214\/aos\/1176347265","article-title":"The jackknife and the bootstrap for general stationary observations","volume":"17","author":"Kunsch","year":"1989","journal-title":"The Annals of Statistics"},{"issue":"1","key":"2024020822013990800_qkad083-B41","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1093\/biomet\/73.1.13","article-title":"Longitudinal data analysis using generalized linear models","volume":"73","author":"Liang","year":"1986","journal-title":"Biometrika"},{"key":"2024020822013990800_qkad083-B42","author":"Lindsay","year":"1988"},{"issue":"1","key":"2024020822013990800_qkad083-B43","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1093\/biomet\/asx079","article-title":"On bias reduction and incidental parameters","volume":"105","author":"Lunardon","year":"2018","journal-title":"Biometrika"},{"issue":"22","key":"2024020822013990800_qkad083-B44","doi-asserted-by":"crossref","first-page":"3596","DOI":"10.1002\/sim.7366","article-title":"Comment on \u2018Small sample GEE estimation of regression parameters for longitudinal data\u2019","volume":"36","author":"Lunardon","year":"2017","journal-title":"Statistics in Medicine"},{"issue":"2","key":"2024020822013990800_qkad083-B45","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/S0304-4076(97)00099-7","article-title":"Approximate bias correction in econometrics","volume":"85","author":"MacKinnon","year":"1998","journal-title":"Journal of Econometrics"},{"issue":"4","key":"2024020822013990800_qkad083-B46","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1093\/aje\/kwx299","article-title":"Separation in logistic regression: Causes, consequences, and control","volume":"187","author":"Mansournia","year":"2018","journal-title":"American Journal of Epidemiology"},{"key":"2024020822013990800_qkad083-B47","doi-asserted-by":"crossref","DOI":"10.1201\/9781351077118","volume-title":"Tensor methods in statistics","author":"McCullagh","year":"2018","edition":"2nd ed."},{"issue":"24","key":"2024020822013990800_qkad083-B48","doi-asserted-by":"crossref","first-page":"615","DOI":"10.21105\/joss.00615","article-title":"Optim: A mathematical optimization package for Julia","volume":"3","author":"Mogensen","year":"2018","journal-title":"Journal of Open Source Software"},{"issue":"1","key":"2024020822013990800_qkad083-B49","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1111\/j.1468-0262.2004.00482.x","article-title":"Higher order properties of GMM and generalized empirical likelihood estimators","volume":"72","author":"Newey","year":"2004","journal-title":"Econometrica"},{"key":"2024020822013990800_qkad083-B50","doi-asserted-by":"crossref","DOI":"10.1142\/3409","volume-title":"Principles of statistical inference from a neo-Fisherian perspective","author":"Pace","year":"1997"},{"issue":"9","key":"2024020822013990800_qkad083-B51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v063.i09","article-title":"Analysis of random fields using CompRandFld","volume":"63","author":"Padoan","year":"2015","journal-title":"Journal of Statistical Software"},{"issue":"489","key":"2024020822013990800_qkad083-B52","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1198\/jasa.2009.tm08577","article-title":"Likelihood-based inference for max-stable processes","volume":"105","author":"Padoan","year":"2010","journal-title":"Journal of the American Statistical Association"},{"issue":"428","key":"2024020822013990800_qkad083-B53","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1080\/01621459.1994.10476870","article-title":"The stationary bootstrap","volume":"89","author":"Politis","year":"1994","journal-title":"Journal of the American Statistical Association"},{"issue":"373","key":"2024020822013990800_qkad083-B54","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1080\/01621459.1981.10477613","article-title":"Concavity of the log likelihood","volume":"76","author":"Pratt","year":"1981","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2024020822013990800_qkad083-B55","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1017\/S0305004100025123","article-title":"Approximate tests of correlation in time-series 3","volume":"45","author":"Quenouille","year":"1949","journal-title":"Mathematical Proceedings of the Cambridge Philosophical Society"},{"issue":"3","key":"2024020822013990800_qkad083-B56","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1093\/biomet\/43.3-4.353","article-title":"Notes on bias in estimation","volume":"43","author":"Quenouille","year":"1956","journal-title":"Biometrika"},{"key":"2024020822013990800_qkad083-B57","author":"R Core Team","year":"2023"},{"key":"2024020822013990800_qkad083-B58","author":"Revels","year":"2016"},{"key":"2024020822013990800_qkad083-B59","author":"Ribeiro","year":"2022"},{"issue":"1","key":"2024020822013990800_qkad083-B60","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1198\/000313002753631330","article-title":"The calculus of M-estimation","volume":"56","author":"Stefanski","year":"2002","journal-title":"The American Statistician"},{"issue":"29","key":"2024020822013990800_qkad083-B61","doi-asserted-by":"crossref","first-page":"14516","DOI":"10.1073\/pnas.1810420116","article-title":"A modern maximum-likelihood theory for high-dimensional logistic regression","volume":"116","author":"Sur","year":"2019","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"2024020822013990800_qkad083-B62","first-page":"12","article-title":"Distribution of informational statistics and a criterion for model fitting [in Japanese]","volume":"153","author":"Takeuchi","year":"1976","journal-title":"Suri-Kagaku"},{"issue":"1","key":"2024020822013990800_qkad083-B63","doi-asserted-by":"crossref","first-page":"2","DOI":"10.4269\/ajtmh.1999.61.2","article-title":"Predicting malaria infection in gambian children from satellite data and bed net use surveys: The importance of spatial correlation in the interpretation of results","volume":"61","author":"Thomson","year":"1999","journal-title":"The American Journal of Tropical Medicine and Hygiene"},{"key":"2024020822013990800_qkad083-B64","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511802256","volume-title":"Asymptotic statistics","author":"van der Vaart","year":"1998"},{"issue":"1","key":"2024020822013990800_qkad083-B65","first-page":"5","article-title":"An overview of composite likelihood methods","volume":"21","author":"Varin","year":"2011","journal-title":"Statistica Sinica"},{"issue":"3","key":"2024020822013990800_qkad083-B66","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1093\/biomet\/92.3.519","article-title":"A note on composite likelihood inference and model selection","volume":"92","author":"Varin","year":"2005","journal-title":"Biometrika"},{"issue":"3","key":"2024020822013990800_qkad083-B67","first-page":"439","article-title":"Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method","volume":"61","author":"Wedderburn","year":"1974","journal-title":"Biometrika"},{"key":"2024020822013990800_qkad083-B68","author":"Wolters","year":"2022"},{"key":"2024020822013990800_qkad083-B69","doi-asserted-by":"crossref","first-page":"24","DOI":"10.3389\/fams.2017.00024","article-title":"Better autologistic regression","volume":"3","author":"Wolters","year":"2017","journal-title":"Frontiers in Applied Mathematics and Statistics"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/86\/1\/62\/56629943\/qkad083.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/86\/1\/62\/56629943\/qkad083.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,26]],"date-time":"2024-10-26T04:58:50Z","timestamp":1729918730000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/86\/1\/62\/7275082"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,16]]},"references-count":69,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,9,16]]},"published-print":{"date-parts":[[2024,2,8]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkad083","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2024,2]]},"published":{"date-parts":[[2023,9,16]]}}}