{"id":"https://openalex.org/W2129286222","doi":"https://doi.org/10.1017/s1748499500000610","title":"Chain-Ladder as Maximum Likelihood Revisited","display_name":"Chain-Ladder as Maximum Likelihood Revisited","publication_year":2009,"publication_date":"2009-03-01","ids":{"openalex":"https://openalex.org/W2129286222","doi":"https://doi.org/10.1017/s1748499500000610","mag":"2129286222"},"language":"en","primary_location":{"id":"doi:10.1017/s1748499500000610","doi":"https://doi.org/10.1017/s1748499500000610","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s1748499500000610","pdf_url":null,"source":{"id":"https://openalex.org/S2764823410","display_name":"Annals of Actuarial Science","issn_l":"1748-4995","issn":["1748-4995","1748-5002"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-c","erih-plus","fnege-4","jufo-1","norway-1","poland-40","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annals of Actuarial Science","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"D. Kuang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4401726840","display_name":"Aon (United Kingdom)","ror":"https://ror.org/052waam31","country_code":"GB","type":"company","lineage":["https://openalex.org/I4401726840"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"D. Kuang","raw_affiliation_strings":["Aon, 8 Devonshire Square, London EC2M 4PL, U.K"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Aon, 8 Devonshire Square, London EC2M 4PL, U.K","institution_ids":["https://openalex.org/I4401726840"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084646163","display_name":"Bent Juhl Nielsen","orcid":"https://orcid.org/0000-0002-1567-4652","observed_orcids":["https://orcid.org/0000-0002-1567-4652"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"B. Nielsen","raw_affiliation_strings":["Nuffield College, Oxford OX1 5AD, UK. :, Email:","Nuffield College, Oxford OX1 5AD, UK. :, Email: bent.nielsen@nuffield.ox.ac.uk"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Nuffield College, Oxford OX1 5AD, UK. :, Email:","institution_ids":[]},{"raw_affiliation_string":"Nuffield College, Oxford OX1 5AD, UK. :, Email: bent.nielsen@nuffield.ox.ac.uk","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085580540","display_name":"Jens Perch Nielsen","orcid":"https://orcid.org/0000-0002-2798-0817","observed_orcids":["https://orcid.org/0000-0002-2798-0817"]},"institutions":[{"id":"https://openalex.org/I180825142","display_name":"City, University of London","ror":"https://ror.org/04489at23","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I180825142","https://openalex.org/I4401726869"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"J. P. Nielsen","raw_affiliation_strings":["Cass Business School , City University London , 106 Bunhill Row , London EC1Y 8TZ , U.K","Nuffield College, Oxford OX1 5AD, UK. :, Email:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Cass Business School , City University London , 106 Bunhill Row , London EC1Y 8TZ , U.K","institution_ids":["https://openalex.org/I180825142"]},{"raw_affiliation_string":"Nuffield College, Oxford OX1 5AD, UK. :, Email:","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5085580540"],"corresponding_institution_ids":["https://openalex.org/I180825142"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":3.6187,"has_fulltext":true,"cited_by_count":39,"citation_normalized_percentile":{"value":0.92591008,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"4","issue":"1","first_page":"105","last_page":"121"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11720","display_name":"Probability and Risk Models","score":0.9121000170707703,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T11720","display_name":"Probability and Risk Models","score":0.9121000170707703,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12011","display_name":"Insurance, Mortality, Demography, Risk Management","score":0.042899999767541885,"subfield":{"id":"https://openalex.org/subfields/3317","display_name":"Demography"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.012299999594688416,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/maximum-likelihood-estimation","display_name":"maximum likelihood estimation","score":0.996999979019165},{"id":"https://openalex.org/keywords/poisson-model","display_name":"Poisson model","score":0.9340000152587891},{"id":"https://openalex.org/keywords/identification-problem","display_name":"identification problem","score":0.6899999976158142},{"id":"https://openalex.org/keywords/boundary-problem","display_name":"boundary problem","score":0.7390000224113464},{"id":"https://openalex.org/keywords/loss-reserving","display_name":"loss reserving","score":0.6639999747276306}],"concepts":[{"id":"https://openalex.org/C49781872","wikidata":"https://www.wikidata.org/wiki/Q1045555","display_name":"Maximum likelihood","level":2,"score":0.746776819229126},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.7462571263313293},{"id":"https://openalex.org/C100906024","wikidata":"https://www.wikidata.org/wiki/Q205692","display_name":"Poisson distribution","level":2,"score":0.602836549282074},{"id":"https://openalex.org/C199185054","wikidata":"https://www.wikidata.org/wiki/Q552299","display_name":"Chain (unit)","level":2,"score":0.597042441368103},{"id":"https://openalex.org/C26503482","wikidata":"https://www.wikidata.org/wiki/Q1898080","display_name":"Underwriting","level":2,"score":0.5688006281852722},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5636817216873169},{"id":"https://openalex.org/C116834253","wikidata":"https://www.wikidata.org/wiki/Q2039217","display_name":"Identification (biology)","level":2,"score":0.5578845739364624},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5552181005477905},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5114027857780457},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4756309986114502},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.4508190453052521},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.44223326444625854},{"id":"https://openalex.org/C191462741","wikidata":"https://www.wikidata.org/wiki/Q6795902","display_name":"Maximum likelihood sequence estimation","level":3,"score":0.4210840165615082},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3558157682418823},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.34335485100746155},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2616526484489441},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.19198453426361084},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1395944058895111},{"id":"https://openalex.org/C162118730","wikidata":"https://www.wikidata.org/wiki/Q1128453","display_name":"Actuarial science","level":1,"score":0.10158106684684753},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08843863010406494},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0},{"id":"https://openalex.org/C59822182","wikidata":"https://www.wikidata.org/wiki/Q441","display_name":"Botany","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1017/s1748499500000610","doi":"https://doi.org/10.1017/s1748499500000610","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s1748499500000610","pdf_url":null,"source":{"id":"https://openalex.org/S2764823410","display_name":"Annals of Actuarial Science","issn_l":"1748-4995","issn":["1748-4995","1748-5002"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-c","erih-plus","fnege-4","jufo-1","norway-1","poland-40","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annals of Actuarial Science","raw_type":"journal-article"},{"id":"pmh:uuid:651a0f26-8665-463a-ae43-0b1a408b5eb7","doi":null,"is_oa":false,"landing_page_url":"http://www.ingentaconnect.com/content/fia/aas/2009/00000004/00000001/art00005","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"http://economics.ouls.ox.ac.uk/14469","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.150.7088","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.7088","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.nuff.ox.ac.uk/economics/papers/2009/w8/ChainLadderML22aug09.pdf","raw_type":"text"},{"id":"mag:2237661274","doi":null,"is_oa":false,"landing_page_url":"https://ideas.repec.org/p/nuf/econwp/0908.html","pdf_url":null,"source":{"id":"https://openalex.org/S4306508693","display_name":"Economics Papers","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Economics Papers","raw_type":null},{"id":"pmh:oai:RePEc:cup:anacsi:v:4:y:2009:i:01:p:105-121_00","doi":null,"is_oa":false,"landing_page_url":"https://www.cambridge.org/core/product/identifier/S1748499500000610/type/journal_article","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":null,"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3930617","display_name":"Econometric analysis of hyperinflation data","funder_award_id":"RES-000-27-0179","funder_id":"https://openalex.org/F4320334630","funder_display_name":"Economic and Social Research Council"},{"id":"https://openalex.org/G841499813","display_name":null,"funder_award_id":"RES-000-","funder_id":"https://openalex.org/F4320334630","funder_display_name":"Economic and Social Research Council"}],"funders":[{"id":"https://openalex.org/F4320334630","display_name":"Economic and Social Research Council","ror":"https://ror.org/03n0ht308"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W608489347","https://openalex.org/W1976422186","https://openalex.org/W1987255543","https://openalex.org/W2030289141","https://openalex.org/W2046336369","https://openalex.org/W2056297376","https://openalex.org/W2088257423","https://openalex.org/W2097536801","https://openalex.org/W2124349276","https://openalex.org/W2152023476","https://openalex.org/W2316942828","https://openalex.org/W2332062191","https://openalex.org/W4231019034","https://openalex.org/W4235809732","https://openalex.org/W4240874411","https://openalex.org/W4249667877","https://openalex.org/W4385628969"],"related_works":["https://openalex.org/W7135644271","https://openalex.org/W3124051089","https://openalex.org/W2316942828","https://openalex.org/W2291984180","https://openalex.org/W1993693796","https://openalex.org/W4416940088","https://openalex.org/W3124926450","https://openalex.org/W1997927915","https://openalex.org/W2934557523","https://openalex.org/W138253319"],"abstract_inverted_index":{"ABSTRACT":[0],"It":[1],"has":[2],"long":[3],"been":[4],"known":[5],"that":[6],"maximum":[7,38],"likelihood":[8,39],"estimation":[9],"in":[10,34,58],"a":[11],"Poisson":[12],"model":[13],"reproduces":[14],"the":[15,30,35,42],"chain-ladder":[16],"technique.":[17],"We":[18],"revisit":[19],"this":[20],"model.":[21],"A":[22],"new":[23],"canonical":[24,43],"parametrisation":[25],"is":[26,68,70],"proposed":[27],"to":[28,50],"circumvent":[29],"inherent":[31],"identification":[32],"problem":[33,54],"parametrisation.":[36],"The":[37,52],"estimators":[40],"for":[41],"parameter":[44],"are":[45],"simple,":[46],"interpretable":[47],"and":[48],"easy":[49],"derive.":[51],"boundary":[53],"where":[55],"all":[56],"observations":[57],"one":[59],"particular":[60,65],"development":[61],"year":[62,67],"or":[63],"on":[64],"underwriting":[66],"zero":[69],"also":[71],"analysed.":[72]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":3}],"updated_date":"2026-10-27T15:51:10.671819","created_date":"2016-06-24T00:00:00"}