{"id":"https://openalex.org/W3142072980","doi":"https://doi.org/10.2307/20075629","title":"The Exact Moments of a Ratio of Quadratic Forms in Normal Variables","display_name":"The Exact Moments of a Ratio of Quadratic Forms in Normal Variables","publication_year":1986,"publication_date":"1986-01-01","ids":{"openalex":"https://openalex.org/W3142072980","doi":"https://doi.org/10.2307/20075629","mag":"3142072980"},"language":"fr","primary_location":{"id":"doi:10.2307/20075629","doi":"https://doi.org/10.2307/20075629","is_oa":false,"landing_page_url":"https://doi.org/10.2307/20075629","pdf_url":null,"source":{"id":"https://openalex.org/S4210170029","display_name":"Annales d \u00c9conomie et de Statistique","issn_l":"0769-489X","issn":["0769-489X","2272-6497"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annales d'\u00c9conomie et de Statistique","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://research.tilburguniversity.edu/en/publications/c6725407-ac3c-44fd-b6d1-55057757c1c0","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075319012","display_name":"Magnus","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Magnus","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5075319012"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.8606,"has_fulltext":false,"cited_by_count":103,"citation_normalized_percentile":{"value":0.88874259,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":99},"biblio":{"volume":"4","issue":"4","first_page":"95","last_page":"95"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.45730000734329224,"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"}},"topics":[{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.45730000734329224,"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"}},{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.17409999668598175,"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"}},{"id":"https://openalex.org/T13141","display_name":"Statistical Methods and Applications","score":0.16979999840259552,"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/exact-moments","display_name":"exact moments","score":0.8889999985694885},{"id":"https://openalex.org/keywords/multivariate-normal-distribution","display_name":"multivariate normal distribution","score":0.7940000295639038},{"id":"https://openalex.org/keywords/mean-vector","display_name":"mean vector","score":0.5149999856948853},{"id":"https://openalex.org/keywords/existence-of-moments","display_name":"existence of moments","score":0.7009999752044678},{"id":"https://openalex.org/keywords/numerical-evaluation-of-integrals","display_name":"numerical evaluation of integrals","score":0.7910000085830688}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6424019932746887},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5524743795394897},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.436867892742157},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3636268079280853},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3245907425880432},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10702487826347351}],"mesh":[],"locations_count":9,"locations":[{"id":"doi:10.2307/20075629","doi":"https://doi.org/10.2307/20075629","is_oa":false,"landing_page_url":"https://doi.org/10.2307/20075629","pdf_url":null,"source":{"id":"https://openalex.org/S4210170029","display_name":"Annales d \u00c9conomie et de Statistique","issn_l":"0769-489X","issn":["0769-489X","2272-6497"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Annales d'\u00c9conomie et de Statistique","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:tiu:tiutis:c6725407-ac3c-44fd-b6d1-55057757c1c0","doi":null,"is_oa":false,"landing_page_url":"https://repository.tilburguniversity.edu/bitstreams/f4f1c52f-76b1-4453-85e4-0e939e9469b4/download","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":"preprint"},{"id":"pmh:oai:tilburguniversity.edu:openaire_cris_publications/c6725407-ac3c-44fd-b6d1-55057757c1c0","doi":null,"is_oa":false,"landing_page_url":"https://repository.tilburguniversity.edu/bitstreams/f4f1c52f-76b1-4453-85e4-0e939e9469b4/download","pdf_url":null,"source":{"id":"https://openalex.org/S4306401490","display_name":"Research portal (Tilburg University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I193700539","host_organization_name":"Tilburg University","host_organization_lineage":["https://openalex.org/I193700539"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Magnus, J R 1986, 'The exact moments of a ratio of quadratic forms in normal variables', Annales d'\u00c9conomie et de Statistique, vol. 4, pp. 95-109.","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:uvt:oai:tilburguniversity.edu:publications/c6725407-ac3c-44fd-b6d1-55057757c1c0","doi":null,"is_oa":true,"landing_page_url":"https://research.tilburguniversity.edu/en/publications/c6725407-ac3c-44fd-b6d1-55057757c1c0","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Annales d'\u00c9conomie et de Statistique, 4, 95 - 109. Association pour le developpement de la recherche en economie et en statistique","raw_type":"info:eu-repo/semantics/article"},{"id":"mag:2259098301","doi":null,"is_oa":false,"landing_page_url":"https://EconPapers.repec.org/RePEc:tiu:tiutis:c6725407-ac3c-44fd-b6d1-55057757c1c0","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":"RePEc: Research Papers in Economics","raw_type":null},{"id":"mag:2345759293","doi":null,"is_oa":false,"landing_page_url":"https://pure.uvt.nl/ws/files/649886/26968_6650.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S2764518860","display_name":"Annals of Economics and Statistics","issn_l":"1968-3863","issn":["1968-3863","2115-4430"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","poland-20","russia-white-list-3","vabb-shw"],"host_organization":"https://openalex.org/P4310311415","host_organization_name":"The National Institute of Statistics and Economic Studies","host_organization_lineage":["https://openalex.org/P4310311415"],"host_organization_lineage_names":["The National Institute of Statistics and Economic Studies"],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Annals of Economics and Statistics","raw_type":null},{"id":"pmh:oai:RePEc:adr:anecst:y:1986:i:4:p:95-109","doi":null,"is_oa":false,"landing_page_url":"http://www.jstor.org/stable/20075629","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"},{"id":"pmh:oai:RePEc:ner:tilbur:urn:nbn:nl:ui:12-153219","doi":null,"is_oa":false,"landing_page_url":"http://arno.uvt.nl/show.cgi?fid=26968","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":"","raw_type":"preprint"},{"id":"mag:3174774938","doi":null,"is_oa":false,"landing_page_url":"https://sticerd.lse.ac.uk/_NEW/PUBLICATIONS/abstract/?index=1180","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:uvt:oai:tilburguniversity.edu:publications/c6725407-ac3c-44fd-b6d1-55057757c1c0","doi":null,"is_oa":true,"landing_page_url":"https://research.tilburguniversity.edu/en/publications/c6725407-ac3c-44fd-b6d1-55057757c1c0","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Annales d'\u00c9conomie et de Statistique, 4, 95 - 109. Association pour le developpement de la recherche en economie et en statistique","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3174774938","https://openalex.org/W2168790274","https://openalex.org/W1534326931","https://openalex.org/W1559583295","https://openalex.org/W1523098808","https://openalex.org/W1991353463","https://openalex.org/W2020964239","https://openalex.org/W2092489900","https://openalex.org/W869284473","https://openalex.org/W2604638194"],"abstract_inverted_index":{"The":[0],"exact":[1],"moments":[2,33,57],"of":[3,55],"x'Ax/x'Bx":[4],"are":[5,58,63],"obtained,":[6],"where":[7],"x":[8],"is":[9,25],"a":[10,44],"normally":[11],"distributed":[12],"vector":[13],"with":[14],"some":[15],"mean":[16],"(possibly":[17],"nonzero)":[18],"and":[19,27],"positive":[20,29],"definite":[21],"covariance":[22],"matrix,":[23],"A":[24],"symmetric":[26],"B":[28],"semi":[30],"definite.":[31],"These":[32],"appear":[34],"as":[35],"simple":[36],"integrals":[37],"which":[38],"can":[39],"be":[40],"evaluated":[41],"numerically":[42],"in":[43],"straightforward":[45],"manner.":[46],"In":[47],"addition,":[48],"the":[49,53,56],"precise":[50],"conditions":[51],"for":[52],"existence":[54],"found.":[59],"Some":[60],"related":[61],"results":[62],"also":[64],"reported.":[65]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2021-04-13T00:00:00"}