{"id":"https://openalex.org/W3038103827","doi":"https://doi.org/10.3390/e22060709","title":"Generalized Entropies, Variance and Applications","display_name":"Generalized Entropies, Variance and Applications","publication_year":2020,"publication_date":"2020-06-26","ids":{"openalex":"https://openalex.org/W3038103827","doi":"https://doi.org/10.3390/e22060709","mag":"3038103827","pmid":"https://pubmed.ncbi.nlm.nih.gov/33286481"},"language":"en","primary_location":{"id":"doi:10.3390/e22060709","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22060709","pdf_url":"https://www.mdpi.com/1099-4300/22/6/709/pdf?version=1593508997","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/1099-4300/22/6/709/pdf?version=1593508997","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085172883","display_name":"Abdolsaeed Toomaj","orcid":"https://orcid.org/0000-0001-8813-473X"},"institutions":[{"id":"https://openalex.org/I4210134932","display_name":"Gonbad Kavous University","ror":"https://ror.org/04a1nf004","country_code":"IR","type":"education","lineage":["https://openalex.org/I4210134932"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Abdolsaeed Toomaj","raw_affiliation_strings":["Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, Basirat Blvd., Shahid Fallahi Street, Gonbad Kavous 4971799151, Golestan Province, Iran"],"raw_orcid":"https://orcid.org/0000-0001-8813-473X","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, Basirat Blvd., Shahid Fallahi Street, Gonbad Kavous 4971799151, Golestan Province, Iran","institution_ids":["https://openalex.org/I4210134932"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053483603","display_name":"Antonio Di Crescenzo","orcid":"https://orcid.org/0000-0003-4751-7341"},"institutions":[{"id":"https://openalex.org/I131729948","display_name":"University of Salerno","ror":"https://ror.org/0192m2k53","country_code":"IT","type":"education","lineage":["https://openalex.org/I131729948"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Antonio Di Crescenzo","raw_affiliation_strings":["Dipartimento di Matematica, Universit\u00e0 degli Studi di Salerno, Via Giovanni Paolo II n. 132, I-84084 Fisciano (SA), Italy"],"raw_orcid":"https://orcid.org/0000-0003-4751-7341","affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00e0 degli Studi di Salerno, Via Giovanni Paolo II n. 132, I-84084 Fisciano (SA), Italy","institution_ids":["https://openalex.org/I131729948"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5053483603"],"corresponding_institution_ids":["https://openalex.org/I131729948"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":3.3237,"has_fulltext":true,"cited_by_count":18,"citation_normalized_percentile":{"value":0.93224709,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"22","issue":"6","first_page":"709","last_page":"709"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9980000257492065,"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":0.9980000257492065,"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/T10780","display_name":"Reliability and Maintenance Optimization","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/2213","display_name":"Safety, Risk, Reliability and Quality"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/residual-entropy","display_name":"Residual entropy","score":0.7938472032546997},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7021617889404297},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.5822713971138},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5513713359832764},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.5472074747085571},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5207183957099915},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5185116529464722},{"id":"https://openalex.org/keywords/cumulative-distribution-function","display_name":"Cumulative distribution function","score":0.5167185664176941},{"id":"https://openalex.org/keywords/poisson-distribution","display_name":"Poisson distribution","score":0.4447077214717865},{"id":"https://openalex.org/keywords/exponential-distribution","display_name":"Exponential distribution","score":0.4298579692840576},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.4285036027431488},{"id":"https://openalex.org/keywords/maximum-entropy-probability-distribution","display_name":"Maximum entropy probability distribution","score":0.4107389748096466},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.35119569301605225},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.3141288161277771},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.19279524683952332},{"id":"https://openalex.org/keywords/probability-density-function","display_name":"Probability density function","score":0.147896409034729},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14615407586097717},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11438792943954468},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.11414298415184021},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.0719282329082489}],"concepts":[{"id":"https://openalex.org/C97785096","wikidata":"https://www.wikidata.org/wiki/Q771478","display_name":"Residual entropy","level":3,"score":0.7938472032546997},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7021617889404297},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.5822713971138},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5513713359832764},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5472074747085571},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5207183957099915},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5185116529464722},{"id":"https://openalex.org/C103784038","wikidata":"https://www.wikidata.org/wiki/Q386228","display_name":"Cumulative distribution function","level":3,"score":0.5167185664176941},{"id":"https://openalex.org/C100906024","wikidata":"https://www.wikidata.org/wiki/Q205692","display_name":"Poisson distribution","level":2,"score":0.4447077214717865},{"id":"https://openalex.org/C55350006","wikidata":"https://www.wikidata.org/wiki/Q237193","display_name":"Exponential distribution","level":2,"score":0.4298579692840576},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4285036027431488},{"id":"https://openalex.org/C60507348","wikidata":"https://www.wikidata.org/wiki/Q6795892","display_name":"Maximum entropy probability distribution","level":3,"score":0.4107389748096466},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.35119569301605225},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.3141288161277771},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.19279524683952332},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.147896409034729},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14615407586097717},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11438792943954468},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.11414298415184021},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0719282329082489},{"id":"https://openalex.org/C124589349","wikidata":"https://www.wikidata.org/wiki/Q3726376","display_name":"Configuration entropy","level":2,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e22060709","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22060709","pdf_url":"https://www.mdpi.com/1099-4300/22/6/709/pdf?version=1593508997","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:33286481","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33286481","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":null},{"id":"pmh:oai:doaj.org/article:1e5b72ffcea24442856c98392c152c3e","is_oa":true,"landing_page_url":"https://doaj.org/article/1e5b72ffcea24442856c98392c152c3e","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 22, Iss 6, p 709 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/22/6/709/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e22060709","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7517246","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7517246","pdf_url":null,"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"}],"best_oa_location":{"id":"doi:10.3390/e22060709","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22060709","pdf_url":"https://www.mdpi.com/1099-4300/22/6/709/pdf?version=1593508997","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/G5195435811","display_name":null,"funder_award_id":"2017JFFHSH","funder_id":"https://openalex.org/F4320321873","funder_display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320321873","display_name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","ror":"https://ror.org/0166hxq48"},{"id":"https://openalex.org/F4320330158","display_name":"Gonbad Kavous University","ror":"https://ror.org/04a1nf004"},{"id":"https://openalex.org/F4320334079","display_name":"Gruppo Nazionale per il Calcolo Scientifico","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3038103827.pdf","grobid_xml":"https://content.openalex.org/works/W3038103827.grobid-xml"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W1526512085","https://openalex.org/W1964588003","https://openalex.org/W1968317750","https://openalex.org/W1970985964","https://openalex.org/W1977702474","https://openalex.org/W1994273251","https://openalex.org/W1994343497","https://openalex.org/W1995875735","https://openalex.org/W2029845722","https://openalex.org/W2034426848","https://openalex.org/W2034485585","https://openalex.org/W2035816347","https://openalex.org/W2036744117","https://openalex.org/W2048832652","https://openalex.org/W2057524960","https://openalex.org/W2066874235","https://openalex.org/W2070761049","https://openalex.org/W2081950515","https://openalex.org/W2087873259","https://openalex.org/W2091470271","https://openalex.org/W2095531255","https://openalex.org/W2101907742","https://openalex.org/W2109345034","https://openalex.org/W2114846044","https://openalex.org/W2116535299","https://openalex.org/W2125644087","https://openalex.org/W2126391818","https://openalex.org/W2139657458","https://openalex.org/W2149259910","https://openalex.org/W2159469674","https://openalex.org/W2313409394","https://openalex.org/W2324402640","https://openalex.org/W2441016763","https://openalex.org/W2465618860","https://openalex.org/W2469410023","https://openalex.org/W2472576221","https://openalex.org/W2710331417","https://openalex.org/W2779665869","https://openalex.org/W2810087923","https://openalex.org/W2947175055","https://openalex.org/W2999844855","https://openalex.org/W3010942683","https://openalex.org/W3038639349","https://openalex.org/W3101292872","https://openalex.org/W3105250080","https://openalex.org/W4245777296","https://openalex.org/W6683542177"],"related_works":["https://openalex.org/W2034921015","https://openalex.org/W3021846434","https://openalex.org/W2064875109","https://openalex.org/W59993211","https://openalex.org/W2999844855","https://openalex.org/W2796583185","https://openalex.org/W1964089340","https://openalex.org/W2324402640","https://openalex.org/W2378001505","https://openalex.org/W2605400039"],"abstract_inverted_index":{"The":[0],"generalized":[1,27,120,130],"cumulative":[2,28,121,131],"residual":[3,29,74,132],"entropy":[4,30],"is":[5,70,124],"a":[6,21,94,106,125],"recently":[7],"defined":[8],"dispersion":[9],"measure.":[10],"In":[11,80],"this":[12,81],"paper,":[13],"we":[14,83],"obtain":[15,115],"some":[16,85],"further":[17],"results":[18,101,117],"for":[19,96,118],"such":[20,60,92],"measure,":[22],"in":[23,89],"relation":[24],"to":[25,108,128],"the":[26,32,46,63,67,73,97,109,119,129],"and":[31,56,76,105],"variance":[33],"of":[34,66,87],"random":[35],"lifetimes.":[36],"We":[37,50,113],"show":[38],"that":[39],"it":[40],"has":[41],"an":[42],"intimate":[43],"connection":[44,107],"with":[45],"non-homogeneous":[47],"Poisson":[48],"process.":[49],"also":[51,114],"get":[52],"new":[53],"expressions,":[54],"bounds":[55],"stochastic":[57],"comparisons":[58],"involving":[59],"measures.":[61],"Moreover,":[62],"dynamic":[64],"version":[65],"mentioned":[68],"notions":[69],"studied":[71],"through":[72],"lifetimes":[75],"suitable":[77],"aging":[78],"notions.":[79],"framework":[82],"achieve":[84],"findings":[86],"interest":[88],"reliability":[90],"theory,":[91],"as":[93],"characterization":[95],"exponential":[98],"distribution,":[99],"various":[100],"on":[102],"k-out-of-n":[103],"systems,":[104],"excess":[110],"wealth":[111],"order.":[112],"similar":[116],"entropy,":[122],"which":[123],"dual":[126],"measure":[127],"entropy.":[133]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":2}],"updated_date":"2026-06-06T09:05:17.133730","created_date":"2025-10-10T00:00:00"}