{"id":"https://openalex.org/W2003746747","doi":"https://doi.org/10.1214/aos/1176345072","title":"Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances","display_name":"Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances","publication_year":1980,"publication_date":"1980-07-01","ids":{"openalex":"https://openalex.org/W2003746747","doi":"https://doi.org/10.1214/aos/1176345072","mag":"2003746747"},"language":"en","primary_location":{"id":"doi:10.1214/aos/1176345072","doi":"https://doi.org/10.1214/aos/1176345072","is_oa":true,"landing_page_url":"https://doi.org/10.1214/aos/1176345072","pdf_url":"https://projecteuclid.org/journals/annals-of-statistics/volume-8/issue-4/Necessary-and-Sufficient-Conditions-for-Explicit-Solutions-in-the-Multivariate/10.1214/aos/1176345072.pdf","source":{"id":"https://openalex.org/S119757635","display_name":"The Annals of Statistics","issn_l":"0090-5364","issn":["0090-5364","2168-8966"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a-star","anvur-class-a","anvur-scientific","cwts-core","dongbi-a","doyens","jufo-3","ki-jl-2","norway-2","poland-200","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Statistics","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://projecteuclid.org/journals/annals-of-statistics/volume-8/issue-4/Necessary-and-Sufficient-Conditions-for-Explicit-Solutions-in-the-Multivariate/10.1214/aos/1176345072.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057552818","display_name":"Ted H. Szatrowski","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Ted H. Szatrowski","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5057552818"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.0959,"has_fulltext":false,"cited_by_count":97,"citation_normalized_percentile":{"value":0.93600517,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"8","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13487","display_name":"Statistical and numerical algorithms","score":0.3488999903202057,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T13487","display_name":"Statistical and numerical algorithms","score":0.3488999903202057,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.21549999713897705,"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":0.20260000228881836,"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":0.8450000286102295},{"id":"https://openalex.org/keywords/multivariate-normal-distribution","display_name":"multivariate normal distribution","score":0.9470000267028809},{"id":"https://openalex.org/keywords/explicit-solutions","display_name":"explicit solutions","score":0.9729999899864197},{"id":"https://openalex.org/keywords/compound-symmetry","display_name":"compound symmetry","score":0.7080000042915344},{"id":"https://openalex.org/keywords/circular-symmetry","display_name":"circular symmetry","score":0.9660000205039978},{"id":"https://openalex.org/keywords/mixed-effects-models","display_name":"mixed models","score":0.8799999952316284},{"id":"https://openalex.org/keywords/asymptotic-distribution","display_name":"asymptotic distribution","score":0.6449999809265137},{"id":"https://openalex.org/keywords/likelihood-ratio-statistic","display_name":"likelihood ratio statistic","score":0.8240000009536743},{"id":"https://openalex.org/keywords/necessary-and-sufficient-conditions","display_name":"necessary and sufficient conditions","score":0.7889999747276306}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9139384031295776},{"id":"https://openalex.org/C61420037","wikidata":"https://www.wikidata.org/wiki/Q7316301","display_name":"Restricted maximum likelihood","level":3,"score":0.7038045525550842},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.6151232719421387},{"id":"https://openalex.org/C49781872","wikidata":"https://www.wikidata.org/wiki/Q1045555","display_name":"Maximum likelihood","level":2,"score":0.606153130531311},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.6038013696670532},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.5577603578567505},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5329936742782593},{"id":"https://openalex.org/C191462741","wikidata":"https://www.wikidata.org/wiki/Q6795902","display_name":"Maximum likelihood sequence estimation","level":3,"score":0.5244648456573486},{"id":"https://openalex.org/C9483764","wikidata":"https://www.wikidata.org/wiki/Q585740","display_name":"Likelihood-ratio test","level":2,"score":0.48899972438812256},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.45524781942367554},{"id":"https://openalex.org/C89106044","wikidata":"https://www.wikidata.org/wiki/Q45284","display_name":"Likelihood function","level":3,"score":0.4172740578651428}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1214/aos/1176345072","doi":"https://doi.org/10.1214/aos/1176345072","is_oa":true,"landing_page_url":"https://doi.org/10.1214/aos/1176345072","pdf_url":"https://projecteuclid.org/journals/annals-of-statistics/volume-8/issue-4/Necessary-and-Sufficient-Conditions-for-Explicit-Solutions-in-the-Multivariate/10.1214/aos/1176345072.pdf","source":{"id":"https://openalex.org/S119757635","display_name":"The Annals of Statistics","issn_l":"0090-5364","issn":["0090-5364","2168-8966"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a-star","anvur-class-a","anvur-scientific","cwts-core","dongbi-a","doyens","jufo-3","ki-jl-2","norway-2","poland-200","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Statistics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.aos/1176345072","doi":null,"is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.aos/1176345072","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1214/aos/1176345072","doi":"https://doi.org/10.1214/aos/1176345072","is_oa":true,"landing_page_url":"https://doi.org/10.1214/aos/1176345072","pdf_url":"https://projecteuclid.org/journals/annals-of-statistics/volume-8/issue-4/Necessary-and-Sufficient-Conditions-for-Explicit-Solutions-in-the-Multivariate/10.1214/aos/1176345072.pdf","source":{"id":"https://openalex.org/S119757635","display_name":"The Annals of Statistics","issn_l":"0090-5364","issn":["0090-5364","2168-8966"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a-star","anvur-class-a","anvur-scientific","cwts-core","dongbi-a","doyens","jufo-3","ki-jl-2","norway-2","poland-200","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Statistics","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.49000000953674316}],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2003746747.pdf","grobid_xml":"https://content.openalex.org/works/W2003746747.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1566825677","https://openalex.org/W2808306767","https://openalex.org/W1547579891","https://openalex.org/W1512911331","https://openalex.org/W1545556469","https://openalex.org/W2965351525","https://openalex.org/W2371672232","https://openalex.org/W2164174361","https://openalex.org/W2886057590","https://openalex.org/W2103185490"],"abstract_inverted_index":{"The":[0],"problem":[1],"of":[2,27,33,40,70,91,98],"finding":[3,71,95],"maximum":[4,57,72,111],"likelihood":[5,58,73,99,112],"estimates":[6,59,74],"for":[7,24,75,94],"patterned":[8],"means":[9],"and":[10,19,30,82,93],"covariance":[11],"matrices":[12],"in":[13,37,51,78,88],"multivariate":[14],"analysis":[15,90],"is":[16],"considered.":[17],"Necessary":[18],"sufficient":[20],"conditions":[21],"are":[22,60],"presented":[23],"the":[25,31,41,55,68,76,79,89,103,106],"existence":[26],"explicit":[28,35,56,110],"solutions":[29,36],"obtaining":[32],"these":[34],"one":[38],"iteration":[39],"scoring":[42],"equations":[43],"from":[44],"any":[45],"positive":[46],"definite":[47],"starting":[48],"point.":[49],"Cases":[50],"which":[52],"averaging":[53],"yields":[54],"discussed.":[61],"These":[62],"results":[63],"can":[64],"be":[65],"applied":[66],"to":[67],"problems":[69],"parameters":[77,104],"complete,":[80],"compound":[81],"circular":[83],"symmetry":[84],"patterns;":[85],"mixed":[86],"models":[87],"variance;":[92],"asymptotic":[96],"distributions":[97],"ratio":[100],"statistics":[101],"when":[102],"under":[105],"null":[107],"hypothesis":[108],"have":[109],"estimates.":[113]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2025-10-10T00:00:00"}