{"id":"https://openalex.org/W3195588398","doi":"https://doi.org/10.1109/lsp.2021.3117443","title":"On the Variability of the Sample Covariance Matrix Under Complex Elliptical Distributions","display_name":"On the Variability of the Sample Covariance Matrix Under Complex Elliptical Distributions","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3195588398","doi":"https://doi.org/10.1109/lsp.2021.3117443","mag":"3195588398"},"language":"en","primary_location":{"id":"doi:10.1109/lsp.2021.3117443","is_oa":true,"landing_page_url":"https://doi.org/10.1109/lsp.2021.3117443","pdf_url":"https://ieeexplore.ieee.org/ielx7/97/4358004/09557837.pdf","source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Letters","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://ieeexplore.ieee.org/ielx7/97/4358004/09557837.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5022025050","display_name":"Elias Raninen","orcid":"https://orcid.org/0000-0001-7461-5399"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Elias Raninen","raw_affiliation_strings":["Department of Signal Processing and Acoustics, Aalto University, 174277 Espoo, Finland, 02150 (e-mail: Elias.raninen@aalto.fi)","Department of Signal Processing and Acoustics, Aalto University, Aalto, Finland#TAB#"],"raw_orcid":"https://orcid.org/0000-0001-7461-5399","affiliations":[{"raw_affiliation_string":"Department of Signal Processing and Acoustics, Aalto University, 174277 Espoo, Finland, 02150 (e-mail: Elias.raninen@aalto.fi)","institution_ids":["https://openalex.org/I9927081"]},{"raw_affiliation_string":"Department of Signal Processing and Acoustics, Aalto University, Aalto, Finland#TAB#","institution_ids":["https://openalex.org/I9927081"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031785324","display_name":"Esa Ollila","orcid":"https://orcid.org/0000-0002-0898-5313"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Esa Ollila","raw_affiliation_strings":["Signal Processing and Acoustics, Aalto-yliopisto, 174277 Espoo, Finland, 02150 (e-mail: esa.ollila@aalto.fi)","Department of Signal Processing and Acoustics, Aalto University, Aalto, Finland#TAB#"],"raw_orcid":"https://orcid.org/0000-0002-0898-5313","affiliations":[{"raw_affiliation_string":"Signal Processing and Acoustics, Aalto-yliopisto, 174277 Espoo, Finland, 02150 (e-mail: esa.ollila@aalto.fi)","institution_ids":["https://openalex.org/I9927081"]},{"raw_affiliation_string":"Department of Signal Processing and Acoustics, Aalto University, Aalto, Finland#TAB#","institution_ids":["https://openalex.org/I9927081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057170468","display_name":"David E. Tyler","orcid":"https://orcid.org/0000-0003-2040-6318"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David Tyler","raw_affiliation_strings":["Department of Statistics, Rutgers The State University of New Jersey, 242612 New Brunswick, New Jersey, United States, (e-mail: dtyler@stat.rutgers.edu)","Department of Statistics, Rutgers, The State University of New Jersey, Piscataway, NJ, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics, Rutgers The State University of New Jersey, 242612 New Brunswick, New Jersey, United States, (e-mail: dtyler@stat.rutgers.edu)","institution_ids":["https://openalex.org/I102322142"]},{"raw_affiliation_string":"Department of Statistics, Rutgers, The State University of New Jersey, Piscataway, NJ, USA","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6429,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.71235278,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"28","issue":null,"first_page":"2092","last_page":"2096"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11871","display_name":"Advanced Statistical Methods and Models","score":0.994700014591217,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.994700014591217,"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9857000112533569,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10243","display_name":"Statistical Methods and Bayesian Inference","score":0.9830999970436096,"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/covariance-matrix","display_name":"Covariance matrix","score":0.7747485637664795},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.741585373878479},{"id":"https://openalex.org/keywords/estimation-of-covariance-matrices","display_name":"Estimation of covariance matrices","score":0.7224461436271667},{"id":"https://openalex.org/keywords/sample-mean-and-sample-covariance","display_name":"Sample mean and sample covariance","score":0.6586434841156006},{"id":"https://openalex.org/keywords/law-of-total-covariance","display_name":"Law of total covariance","score":0.6488234400749207},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.6150869131088257},{"id":"https://openalex.org/keywords/scatter-matrix","display_name":"Scatter matrix","score":0.5775898098945618},{"id":"https://openalex.org/keywords/elliptical-distribution","display_name":"Elliptical distribution","score":0.5697827339172363},{"id":"https://openalex.org/keywords/equivariant-map","display_name":"Equivariant map","score":0.5437454581260681},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.5323052406311035},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4799690246582031},{"id":"https://openalex.org/keywords/rational-quadratic-covariance-function","display_name":"Rational quadratic covariance function","score":0.4715478718280792},{"id":"https://openalex.org/keywords/covariance-intersection","display_name":"Covariance intersection","score":0.4563773572444916},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4446914792060852},{"id":"https://openalex.org/keywords/covariance-mapping","display_name":"Covariance mapping","score":0.4364203214645386},{"id":"https://openalex.org/keywords/covariance-function","display_name":"Covariance function","score":0.4111120402812958},{"id":"https://openalex.org/keywords/wishart-distribution","display_name":"Wishart distribution","score":0.14973357319831848},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1344958245754242}],"concepts":[{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.7747485637664795},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.741585373878479},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.7224461436271667},{"id":"https://openalex.org/C181243257","wikidata":"https://www.wikidata.org/wiki/Q1693522","display_name":"Sample mean and sample covariance","level":3,"score":0.6586434841156006},{"id":"https://openalex.org/C126372606","wikidata":"https://www.wikidata.org/wiki/Q6503511","display_name":"Law of total covariance","level":5,"score":0.6488234400749207},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.6150869131088257},{"id":"https://openalex.org/C176917957","wikidata":"https://www.wikidata.org/wiki/Q7430596","display_name":"Scatter matrix","level":4,"score":0.5775898098945618},{"id":"https://openalex.org/C130545031","wikidata":"https://www.wikidata.org/wiki/Q5365806","display_name":"Elliptical distribution","level":4,"score":0.5697827339172363},{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.5437454581260681},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.5323052406311035},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4799690246582031},{"id":"https://openalex.org/C148893098","wikidata":"https://www.wikidata.org/wiki/Q7295778","display_name":"Rational quadratic covariance function","level":5,"score":0.4715478718280792},{"id":"https://openalex.org/C83042196","wikidata":"https://www.wikidata.org/wiki/Q5178898","display_name":"Covariance intersection","level":4,"score":0.4563773572444916},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4446914792060852},{"id":"https://openalex.org/C129759605","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance mapping","level":5,"score":0.4364203214645386},{"id":"https://openalex.org/C137250428","wikidata":"https://www.wikidata.org/wiki/Q5178897","display_name":"Covariance function","level":3,"score":0.4111120402812958},{"id":"https://openalex.org/C33962027","wikidata":"https://www.wikidata.org/wiki/Q1930697","display_name":"Wishart distribution","level":3,"score":0.14973357319831848},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1344958245754242},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/lsp.2021.3117443","is_oa":true,"landing_page_url":"https://doi.org/10.1109/lsp.2021.3117443","pdf_url":"https://ieeexplore.ieee.org/ielx7/97/4358004/09557837.pdf","source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Letters","raw_type":"journal-article"},{"id":"mag:3195588398","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2108.08047","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:aaltodoc.aalto.fi:123456789/111190","is_oa":true,"landing_page_url":"https://research.aalto.fi/en/publications/4bcc687c-6a81-460e-a52b-396f5bdfaa05","pdf_url":null,"source":{"id":"https://openalex.org/S4306401662","display_name":"Aaltodoc (Aalto University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I9927081","host_organization_name":"Aalto University","host_organization_lineage":["https://openalex.org/I9927081"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.2108.08047","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2108.08047","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1109/lsp.2021.3117443","is_oa":true,"landing_page_url":"https://doi.org/10.1109/lsp.2021.3117443","pdf_url":"https://ieeexplore.ieee.org/ielx7/97/4358004/09557837.pdf","source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Letters","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4430026858","display_name":null,"funder_award_id":"DMS-1812198","funder_id":"https://openalex.org/F4320335353","funder_display_name":"National Science Foundation of Sri Lanka"}],"funders":[{"id":"https://openalex.org/F4320335353","display_name":"National Science Foundation of Sri Lanka","ror":"https://ror.org/010xaa060"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3195588398.pdf","grobid_xml":"https://content.openalex.org/works/W3195588398.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W184583885","https://openalex.org/W1533423434","https://openalex.org/W1973176871","https://openalex.org/W1986921156","https://openalex.org/W1992511687","https://openalex.org/W1993351044","https://openalex.org/W2001994691","https://openalex.org/W2025907885","https://openalex.org/W2034783362","https://openalex.org/W2037095848","https://openalex.org/W2042292499","https://openalex.org/W2056238226","https://openalex.org/W2061811442","https://openalex.org/W2062052013","https://openalex.org/W2062125287","https://openalex.org/W2066218102","https://openalex.org/W2098918476","https://openalex.org/W2117454350","https://openalex.org/W2119963163","https://openalex.org/W2144158572","https://openalex.org/W2167430018","https://openalex.org/W2888860102","https://openalex.org/W2950381474","https://openalex.org/W3111439133","https://openalex.org/W3124158341","https://openalex.org/W3124979234"],"related_works":["https://openalex.org/W3208495656","https://openalex.org/W1971120007","https://openalex.org/W60246654","https://openalex.org/W2060816402","https://openalex.org/W3048436498","https://openalex.org/W2765709781","https://openalex.org/W975297808","https://openalex.org/W2398418623","https://openalex.org/W1650242368","https://openalex.org/W2103482025","https://openalex.org/W3138564999","https://openalex.org/W2049438561","https://openalex.org/W3048841446","https://openalex.org/W2768781802","https://openalex.org/W3014694112","https://openalex.org/W1807129068","https://openalex.org/W2154994549","https://openalex.org/W2030376515","https://openalex.org/W3101788651","https://openalex.org/W2148213881"],"abstract_inverted_index":{"We":[0,20],"derive":[1,26],"the":[2,5,27,31,54],"form":[3],"of":[4,30,53],"variance-covariance":[6,28],"matrix":[7,29,34],"for":[8],"any":[9],"affine":[10],"equivariant":[11],"matrix-valued":[12],"statistics":[13],"when":[14,45],"sampling":[15],"from":[16],"complex":[17],"elliptical":[18],"distributions.":[19],"then":[21],"use":[22],"this":[23],"result":[24],"to":[25],"sample":[32],"covariance":[33],"(SCM)":[35],"as":[36,38],"well":[37],"its":[39],"theoretical":[40],"mean":[41],"squared":[42],"error":[43],"(MSE)":[44],"finite":[46],"fourth-order":[47],"moments":[48],"exist.":[49],"Finally,":[50],"illustrative":[51],"examples":[52],"formulas":[55],"are":[56],"presented.":[57]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2025-10-10T00:00:00"}