{"id":"https:\/\/openalex.org\/W4389144950","doi":"https:\/\/doi.org\/10.3390\/e25121598","title":"Bridging Extremes: The Invertible Bimodal Gumbel Distribution","display_name":"Bridging Extremes: The Invertible Bimodal Gumbel Distribution","publication_year":2023,"publication_date":"2023-11-29","ids":{"openalex":"https:\/\/openalex.org\/W4389144950","doi":"https:\/\/doi.org\/10.3390\/e25121598","pmid":"https:\/\/pubmed.ncbi.nlm.nih.gov\/38136478"},"language":"en","primary_location":{"id":"doi:10.3390\/e25121598","is_oa":true,"landing_page_url":"https:\/\/doi.org\/10.3390\/e25121598","pdf_url":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1598\/pdf?version=1701221251","source":{"id":"https:\/\/openalex.org\/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https:\/\/openalex.org\/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https:\/\/openalex.org\/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https:\/\/openalex.org\/licenses\/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1598\/pdf?version=1701221251","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https:\/\/openalex.org\/A5045144614","display_name":"Cira E. G. Otiniano","orcid":"https:\/\/orcid.org\/0000-0002-5619-0478"},"institutions":[{"id":"https:\/\/openalex.org\/I150729083","display_name":"Universidade de Bras\u00edlia","ror":"https:\/\/ror.org\/02xfp8v59","country_code":"BR","type":"education","lineage":["https:\/\/openalex.org\/I150729083"]}],"countries":["BR"],"is_corresponding":true,"raw_author_name":"Cira G. Otiniano","raw_affiliation_strings":["Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil"],"raw_orcid":"https:\/\/orcid.org\/0000-0002-5619-0478","affiliations":[{"raw_affiliation_string":"Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil","institution_ids":["https:\/\/openalex.org\/I150729083"]}]},{"author_position":"middle","author":{"id":"https:\/\/openalex.org\/A5113058145","display_name":"Eduarda B. Silva","orcid":null},"institutions":[{"id":"https:\/\/openalex.org\/I150729083","display_name":"Universidade de Bras\u00edlia","ror":"https:\/\/ror.org\/02xfp8v59","country_code":"BR","type":"education","lineage":["https:\/\/openalex.org\/I150729083"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Eduarda B. Silva","raw_affiliation_strings":["Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil","institution_ids":["https:\/\/openalex.org\/I150729083"]}]},{"author_position":"middle","author":{"id":"https:\/\/openalex.org\/A5058633744","display_name":"Raul Matsushita","orcid":"https:\/\/orcid.org\/0000-0001-8864-6356"},"institutions":[{"id":"https:\/\/openalex.org\/I150729083","display_name":"Universidade de Bras\u00edlia","ror":"https:\/\/ror.org\/02xfp8v59","country_code":"BR","type":"education","lineage":["https:\/\/openalex.org\/I150729083"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Raul Y. Matsushita","raw_affiliation_strings":["Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil"],"raw_orcid":"https:\/\/orcid.org\/0000-0001-8864-6356","affiliations":[{"raw_affiliation_string":"Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil","institution_ids":["https:\/\/openalex.org\/I150729083"]}]},{"author_position":"last","author":{"id":"https:\/\/openalex.org\/A5109295615","display_name":"Alan Silva","orcid":"https:\/\/orcid.org\/0009-0006-3084-3801"},"institutions":[{"id":"https:\/\/openalex.org\/I17974374","display_name":"Universidade de S\u00e3o Paulo","ror":"https:\/\/ror.org\/036rp1748","country_code":"BR","type":"education","lineage":["https:\/\/openalex.org\/I17974374"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Alan Silva","raw_affiliation_strings":["Institute of Mathematics and Statistics, S\u00e3o Paulo University, S\u00e3o Paulo 05508-220, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics and Statistics, S\u00e3o Paulo University, S\u00e3o Paulo 05508-220, Brazil","institution_ids":["https:\/\/openalex.org\/I17974374"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https:\/\/openalex.org\/A5045144614"],"corresponding_institution_ids":["https:\/\/openalex.org\/I150729083"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2224},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2224},"fwci":0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.09665707,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"25","issue":"12","first_page":"1598","last_page":"1598"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https:\/\/openalex.org\/T11186","display_name":"Hydrology and Drought Analysis","score":0.9987999796867371,"subfield":{"id":"https:\/\/openalex.org\/subfields\/2306","display_name":"Global and Planetary Change"},"field":{"id":"https:\/\/openalex.org\/fields\/23","display_name":"Environmental Science"},"domain":{"id":"https:\/\/openalex.org\/domains\/3","display_name":"Physical Sciences"}},"topics":[{"id":"https:\/\/openalex.org\/T11186","display_name":"Hydrology and Drought Analysis","score":0.9987999796867371,"subfield":{"id":"https:\/\/openalex.org\/subfields\/2306","display_name":"Global and Planetary Change"},"field":{"id":"https:\/\/openalex.org\/fields\/23","display_name":"Environmental Science"},"domain":{"id":"https:\/\/openalex.org\/domains\/3","display_name":"Physical Sciences"}},{"id":"https:\/\/openalex.org\/T10282","display_name":"Financial Risk and Volatility Modeling","score":0.9977999925613403,"subfield":{"id":"https:\/\/openalex.org\/subfields\/2003","display_name":"Finance"},"field":{"id":"https:\/\/openalex.org\/fields\/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https:\/\/openalex.org\/domains\/2","display_name":"Social Sciences"}},{"id":"https:\/\/openalex.org\/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9937000274658203,"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\/gumbel-distribution","display_name":"Gumbel distribution","score":0.9882591962814331},{"id":"https:\/\/openalex.org\/keywords\/invertible-matrix","display_name":"Invertible matrix","score":0.8099955916404724},{"id":"https:\/\/openalex.org\/keywords\/generalized-extreme-value-distribution","display_name":"Generalized extreme value distribution","score":0.5448763370513916},{"id":"https:\/\/openalex.org\/keywords\/cumulative-distribution-function","display_name":"Cumulative distribution function","score":0.5280457735061646},{"id":"https:\/\/openalex.org\/keywords\/extreme-value-theory","display_name":"Extreme value theory","score":0.5269830226898193},{"id":"https:\/\/openalex.org\/keywords\/mathematics","display_name":"Mathematics","score":0.5209009647369385},{"id":"https:\/\/openalex.org\/keywords\/distribution","display_name":"Distribution (mathematics)","score":0.4746984839439392},{"id":"https:\/\/openalex.org\/keywords\/applied-mathematics","display_name":"Applied mathematics","score":0.45531779527664185},{"id":"https:\/\/openalex.org\/keywords\/statistical-physics","display_name":"Statistical physics","score":0.35140496492385864},{"id":"https:\/\/openalex.org\/keywords\/computer-science","display_name":"Computer science","score":0.3475087881088257},{"id":"https:\/\/openalex.org\/keywords\/econometrics","display_name":"Econometrics","score":0.32615742087364197},{"id":"https:\/\/openalex.org\/keywords\/statistics","display_name":"Statistics","score":0.2646823227405548},{"id":"https:\/\/openalex.org\/keywords\/probability-density-function","display_name":"Probability density function","score":0.253194659948349},{"id":"https:\/\/openalex.org\/keywords\/mathematical-analysis","display_name":"Mathematical analysis","score":0.1771872341632843},{"id":"https:\/\/openalex.org\/keywords\/pure-mathematics","display_name":"Pure mathematics","score":0.15548741817474365},{"id":"https:\/\/openalex.org\/keywords\/physics","display_name":"Physics","score":0.10011875629425049}],"concepts":[{"id":"https:\/\/openalex.org\/C137610916","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q1096862","display_name":"Gumbel distribution","level":3,"score":0.9882591962814331},{"id":"https:\/\/openalex.org\/C96442724","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q242188","display_name":"Invertible matrix","level":2,"score":0.8099955916404724},{"id":"https:\/\/openalex.org\/C169707849","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q1617240","display_name":"Generalized extreme value distribution","level":3,"score":0.5448763370513916},{"id":"https:\/\/openalex.org\/C103784038","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q386228","display_name":"Cumulative distribution function","level":3,"score":0.5280457735061646},{"id":"https:\/\/openalex.org\/C147581598","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q729429","display_name":"Extreme value theory","level":2,"score":0.5269830226898193},{"id":"https:\/\/openalex.org\/C33923547","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q395","display_name":"Mathematics","level":0,"score":0.5209009647369385},{"id":"https:\/\/openalex.org\/C110121322","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.4746984839439392},{"id":"https:\/\/openalex.org\/C28826006","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q33521","display_name":"Applied mathematics","level":1,"score":0.45531779527664185},{"id":"https:\/\/openalex.org\/C121864883","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q677916","display_name":"Statistical physics","level":1,"score":0.35140496492385864},{"id":"https:\/\/openalex.org\/C41008148","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q21198","display_name":"Computer science","level":0,"score":0.3475087881088257},{"id":"https:\/\/openalex.org\/C149782125","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q160039","display_name":"Econometrics","level":1,"score":0.32615742087364197},{"id":"https:\/\/openalex.org\/C105795698","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q12483","display_name":"Statistics","level":1,"score":0.2646823227405548},{"id":"https:\/\/openalex.org\/C197055811","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q207522","display_name":"Probability density function","level":2,"score":0.253194659948349},{"id":"https:\/\/openalex.org\/C134306372","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1771872341632843},{"id":"https:\/\/openalex.org\/C202444582","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q837863","display_name":"Pure mathematics","level":1,"score":0.15548741817474365},{"id":"https:\/\/openalex.org\/C121332964","wikidata":"https:\/\/www.wikidata.org\/wiki\/Q413","display_name":"Physics","level":0,"score":0.10011875629425049}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390\/e25121598","is_oa":true,"landing_page_url":"https:\/\/doi.org\/10.3390\/e25121598","pdf_url":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1598\/pdf?version=1701221251","source":{"id":"https:\/\/openalex.org\/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https:\/\/openalex.org\/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https:\/\/openalex.org\/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https:\/\/openalex.org\/licenses\/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:38136478","is_oa":false,"landing_page_url":"https:\/\/pubmed.ncbi.nlm.nih.gov\/38136478","pdf_url":null,"source":{"id":"https:\/\/openalex.org\/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https:\/\/openalex.org\/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https:\/\/openalex.org\/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":"Journal Article"},{"id":"pmh:oai:pubmedcentral.nih.gov:10742741","is_oa":true,"landing_page_url":"https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/10742741","pdf_url":"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC10742741\/pdf\/entropy-25-01598.pdf","source":{"id":"https:\/\/openalex.org\/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https:\/\/openalex.org\/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https:\/\/openalex.org\/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"},{"id":"pmh:oai:doaj.org\/article:eaea3937dc35497d920b503c543bb2f3","is_oa":true,"landing_page_url":"https:\/\/doaj.org\/article\/eaea3937dc35497d920b503c543bb2f3","pdf_url":null,"source":{"id":"https:\/\/openalex.org\/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https:\/\/openalex.org\/licenses\/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 25, Iss 12, p 1598 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390\/e25121598","is_oa":true,"landing_page_url":"https:\/\/doi.org\/10.3390\/e25121598","pdf_url":"https:\/\/www.mdpi.com\/1099-4300\/25\/12\/1598\/pdf?version=1701221251","source":{"id":"https:\/\/openalex.org\/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https:\/\/openalex.org\/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https:\/\/openalex.org\/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https:\/\/openalex.org\/licenses\/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https:\/\/openalex.org\/G1131692237","display_name":null,"funder_award_id":"311548\/2022-9)","funder_id":"https:\/\/openalex.org\/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https:\/\/openalex.org\/G2634101342","display_name":null,"funder_award_id":"311548\/2022-9","funder_id":"https:\/\/openalex.org\/F4320321091","funder_display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior"},{"id":"https:\/\/openalex.org\/G29316538","display_name":null,"funder_award_id":"(Code 001)","funder_id":"https:\/\/openalex.org\/F4320321091","funder_display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior"},{"id":"https:\/\/openalex.org\/G3535238603","display_name":null,"funder_award_id":"00193-00001860\/2023-17","funder_id":"https:\/\/openalex.org\/F4320323350","funder_display_name":"Funda\u00e7\u00e3o de Apoio \u00e0 Pesquisa do Distrito Federal"},{"id":"https:\/\/openalex.org\/G4063591082","display_name":null,"funder_award_id":"00193-00001860\/2023-17","funder_id":"https:\/\/openalex.org\/F4320321091","funder_display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior"},{"id":"https:\/\/openalex.org\/G552760565","display_name":null,"funder_award_id":"00193","funder_id":"https:\/\/openalex.org\/F4320321091","funder_display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior"},{"id":"https:\/\/openalex.org\/G6756993358","display_name":null,"funder_award_id":"\/2022-9","funder_id":"https:\/\/openalex.org\/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https:\/\/openalex.org\/G6881636641","display_name":null,"funder_award_id":"Code 001","funder_id":"https:\/\/openalex.org\/F4320323350","funder_display_name":"Funda\u00e7\u00e3o de Apoio \u00e0 Pesquisa do Distrito Federal"},{"id":"https:\/\/openalex.org\/G8285809817","display_name":null,"funder_award_id":"00193","funder_id":"https:\/\/openalex.org\/F4320323350","funder_display_name":"Funda\u00e7\u00e3o de Apoio \u00e0 Pesquisa do Distrito Federal"}],"funders":[{"id":"https:\/\/openalex.org\/F4320321091","display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","ror":"https:\/\/ror.org\/00x0ma614"},{"id":"https:\/\/openalex.org\/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https:\/\/ror.org\/03swz6y49"},{"id":"https:\/\/openalex.org\/F4320323350","display_name":"Funda\u00e7\u00e3o de Apoio \u00e0 Pesquisa do Distrito Federal","ror":"https:\/\/ror.org\/04djvx395"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https:\/\/content.openalex.org\/works\/W4389144950.pdf","grobid_xml":"https:\/\/content.openalex.org\/works\/W4389144950.grobid-xml"},"referenced_works_count":24,"referenced_works":["https:\/\/openalex.org\/W224229492","https:\/\/openalex.org\/W419547886","https:\/\/openalex.org\/W1966706927","https:\/\/openalex.org\/W1980437132","https:\/\/openalex.org\/W2004044457","https:\/\/openalex.org\/W2008490700","https:\/\/openalex.org\/W2065374531","https:\/\/openalex.org\/W2077613340","https:\/\/openalex.org\/W2095075780","https:\/\/openalex.org\/W2122988071","https:\/\/openalex.org\/W2139224138","https:\/\/openalex.org\/W2152278349","https:\/\/openalex.org\/W2152552334","https:\/\/openalex.org\/W2186238461","https:\/\/openalex.org\/W2503932823","https:\/\/openalex.org\/W2592905339","https:\/\/openalex.org\/W2769855593","https:\/\/openalex.org\/W2950627632","https:\/\/openalex.org\/W3026042799","https:\/\/openalex.org\/W3083220658","https:\/\/openalex.org\/W3177092674","https:\/\/openalex.org\/W3204683910","https:\/\/openalex.org\/W6608846882","https:\/\/openalex.org\/W6746121116"],"related_works":["https:\/\/openalex.org\/W2750430821","https:\/\/openalex.org\/W2950895695","https:\/\/openalex.org\/W2951988075","https:\/\/openalex.org\/W2592905339","https:\/\/openalex.org\/W163470745","https:\/\/openalex.org\/W2736068589","https:\/\/openalex.org\/W2167010909","https:\/\/openalex.org\/W1862346859","https:\/\/openalex.org\/W2368561115","https:\/\/openalex.org\/W4210771670"],"abstract_inverted_index":{"This":[0,61],"paper":[1,62],"introduces":[2],"a":[3,14,49],"novel":[4],"three-parameter":[5],"invertible":[6,70],"bimodal":[7,38,71],"Gumbel":[8,39,72],"distribution,":[9],"addressing":[10],"the":[11,43,64,69],"need":[12],"for":[13],"versatile":[15],"statistical":[16],"tool":[17],"capable":[18],"of":[19,68,82],"simultaneously":[20],"modeling":[21],"maximum":[22,101],"and":[23,34,59,66,80,102],"minimum":[24,103],"extremes":[25],"in":[26,42],"various":[27],"fields":[28],"such":[29],"as":[30],"hydrology,":[31],"meteorology,":[32],"finance,":[33],"insurance.":[35],"Unlike":[36],"previous":[37],"distributions":[40],"available":[41],"literature,":[44],"our":[45,97],"proposed":[46],"model":[47],"features":[48],"simple":[50],"closed-form":[51],"cumulative":[52],"distribution":[53,73],"function,":[54],"enhancing":[55],"its":[56],"computational":[57],"attractiveness":[58],"applicability.":[60],"elucidates":[63],"behavior":[65],"advantages":[67],"through":[74],"detailed":[75],"mathematical":[76],"formulations,":[77],"graphical":[78],"illustrations,":[79],"exploration":[81],"distributional":[83],"characteristics.":[84],"We":[85],"illustrate":[86],"using":[87],"financial":[88],"data":[89],"to":[90],"estimate":[91],"Value":[92],"at":[93],"Risk":[94],"(VaR)":[95],"from":[96],"suggested":[98],"model,":[99],"considering":[100],"blocks":[104],"simultaneously.":[105]},"counts_by_year":[],"updated_date":"2026-08-28T12:50:07.497085","created_date":"2025-10-10T00:00:00"}