{"id":"https://openalex.org/W2095196337","doi":"https://doi.org/10.1155/2008/382948","title":"Order Statistics and Benford\u2032s Law","display_name":"Order Statistics and Benford\u2032s Law","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2095196337","doi":"https://doi.org/10.1155/2008/382948","mag":"2095196337"},"language":"en","primary_location":{"id":"doi:10.1155/2008/382948","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2008/382948","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2008/382948.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/ijmms/2008/382948.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Steven J. Miller","orcid":null},"institutions":[{"id":"https://openalex.org/I22675022","display_name":"Williams College","ror":"https://ror.org/04avkmd49","country_code":"US","type":"education","lineage":["https://openalex.org/I22675022"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Steven J. Miller","raw_affiliation_strings":["Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA","institution_ids":["https://openalex.org/I22675022"]}]},{"author_position":"last","author":{"id":null,"display_name":"Mark J. Nigrini","orcid":null},"institutions":[{"id":"https://openalex.org/I64281891","display_name":"College of New Jersey","ror":"https://ror.org/02nx5r318","country_code":"US","type":"education","lineage":["https://openalex.org/I64281891"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mark J. Nigrini","raw_affiliation_strings":["Accounting and Information Systems, School of Business, The College of New Jersey, Ewing, NJ 08628, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Accounting and Information Systems, School of Business, The College of New Jersey, Ewing, NJ 08628, USA","institution_ids":["https://openalex.org/I64281891"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.6088,"has_fulltext":true,"cited_by_count":25,"citation_normalized_percentile":{"value":0.72696778,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"2008","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.9979000091552734,"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/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.9979000091552734,"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/T11716","display_name":"Random Matrices and Applications","score":0.00019999999494757503,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/order-statistic","display_name":"Order statistic","score":0.6837000250816345},{"id":"https://openalex.org/keywords/natural-exponential-family","display_name":"Natural exponential family","score":0.6047000288963318},{"id":"https://openalex.org/keywords/exponential-distribution","display_name":"Exponential distribution","score":0.5839999914169312},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5799999833106995},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5685999989509583},{"id":"https://openalex.org/keywords/heavy-tailed-distribution","display_name":"Heavy-tailed distribution","score":0.5562000274658203},{"id":"https://openalex.org/keywords/pareto-distribution","display_name":"Pareto distribution","score":0.5504999756813049},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5281000137329102},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.4641000032424927}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8981999754905701},{"id":"https://openalex.org/C44082924","wikidata":"https://www.wikidata.org/wiki/Q1767128","display_name":"Order statistic","level":2,"score":0.6837000250816345},{"id":"https://openalex.org/C60775368","wikidata":"https://www.wikidata.org/wiki/Q17099337","display_name":"Natural exponential family","level":3,"score":0.6047000288963318},{"id":"https://openalex.org/C55350006","wikidata":"https://www.wikidata.org/wiki/Q237193","display_name":"Exponential distribution","level":2,"score":0.5839999914169312},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5799999833106995},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5685999989509583},{"id":"https://openalex.org/C1602530","wikidata":"https://www.wikidata.org/wiki/Q1472677","display_name":"Heavy-tailed distribution","level":3,"score":0.5562000274658203},{"id":"https://openalex.org/C190373308","wikidata":"https://www.wikidata.org/wiki/Q837683","display_name":"Pareto distribution","level":2,"score":0.5504999756813049},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5281000137329102},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.4641000032424927},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4498000144958496},{"id":"https://openalex.org/C149717495","wikidata":"https://www.wikidata.org/wiki/Q117806","display_name":"Gamma distribution","level":2,"score":0.4253999888896942},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.41620001196861267},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.39629998803138733},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3790000081062317},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.3765000104904175},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.3736000061035156},{"id":"https://openalex.org/C109045618","wikidata":"https://www.wikidata.org/wiki/Q5421535","display_name":"Exponentially modified Gaussian distribution","level":4,"score":0.367000013589859},{"id":"https://openalex.org/C55974624","wikidata":"https://www.wikidata.org/wiki/Q1188504","display_name":"Exponential family","level":2,"score":0.3643999993801117},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.34880000352859497},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.33629998564720154},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3287999927997589},{"id":"https://openalex.org/C108882938","wikidata":"https://www.wikidata.org/wiki/Q274506","display_name":"Uniform distribution (continuous)","level":2,"score":0.32820001244544983},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.31850001215934753},{"id":"https://openalex.org/C180188523","wikidata":"https://www.wikidata.org/wiki/Q574576","display_name":"Exponential decay","level":2,"score":0.2985000014305115},{"id":"https://openalex.org/C49016629","wikidata":"https://www.wikidata.org/wiki/Q3075235","display_name":"Double exponential function","level":3,"score":0.2962000072002411},{"id":"https://openalex.org/C57205106","wikidata":"https://www.wikidata.org/wiki/Q3258519","display_name":"Inverse-chi-squared distribution","level":4,"score":0.29409998655319214},{"id":"https://openalex.org/C121628295","wikidata":"https://www.wikidata.org/wiki/Q5532428","display_name":"Generalized beta distribution","level":4,"score":0.2921999990940094},{"id":"https://openalex.org/C183057437","wikidata":"https://www.wikidata.org/wiki/Q671617","display_name":"Laplace distribution","level":3,"score":0.2791999876499176},{"id":"https://openalex.org/C199879729","wikidata":"https://www.wikidata.org/wiki/Q205140","display_name":"Infinite set","level":3,"score":0.26489999890327454},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.2531999945640564}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1155/2008/382948","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2008/382948","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2008/382948.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:math/0601344","is_oa":true,"landing_page_url":"http://arxiv.org/abs/math/0601344","pdf_url":"https://arxiv.org/pdf/math/0601344","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:1d4116c9ddae4f01ace2a30b19a63f53","is_oa":true,"landing_page_url":"https://doaj.org/article/1d4116c9ddae4f01ace2a30b19a63f53","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":"International Journal of Mathematics and Mathematical Sciences, Vol 2008 (2008)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2008/382948","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2008/382948","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2008/382948.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5406829959","display_name":"Investigations on Low-Lying Zeros of L-Functions","funder_award_id":"0600848","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8749057747","display_name":null,"funder_award_id":"DMS-0600848","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2095196337.pdf","grobid_xml":"https://content.openalex.org/works/W2095196337.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W53181715","https://openalex.org/W124167277","https://openalex.org/W1481984145","https://openalex.org/W1535089941","https://openalex.org/W1537994955","https://openalex.org/W1644793032","https://openalex.org/W1987331682","https://openalex.org/W1998432070","https://openalex.org/W1998686162","https://openalex.org/W2025658512","https://openalex.org/W2058065576","https://openalex.org/W2089017729","https://openalex.org/W2100369046","https://openalex.org/W2110835021","https://openalex.org/W2125173025","https://openalex.org/W2504767107","https://openalex.org/W4229711531","https://openalex.org/W4233034123","https://openalex.org/W4246315805","https://openalex.org/W4247250173","https://openalex.org/W6629021196"],"related_works":[],"abstract_inverted_index":{"Fix":[0],"a":[1,34,113,125],"base":[2,20,66],"B":[3,21],"&gt;":[4],"1":[5,70,148],"and":[6,121,124,144,193],"let":[7],"\u03b6":[8,19,56],"have":[9],"the":[10,14,38,45,51,78,89,92,95,134,152,163,171,174,178,191,196],"standard":[11],"exponential":[12,64,104,157,186],"distribution;":[13],"distribution":[15,39,90,116,135,172,198],"of":[16,18,40,42,47,55,91,94,136,138,151,173,176],"digits":[17,41,93,137,175],"is":[22,50],"known":[23],"to":[24,28,102,182,200],"be":[25,75],"very":[26],"close":[27],"Benford\u2032s":[29,183],"law.":[30],"If":[31,77,106],"there":[32],"exists":[33],"C":[35,43],"such":[36],"that":[37,54,60,195],"times":[44],"elements":[46],"some":[48],"set":[49,61],"same":[52],"as":[53,85],",":[57,71],"we":[58],"say":[59],"exhibits":[62],"shifted":[63,103,156,185],"behavior":[65],"B.":[67],"Let":[68],"X":[69,73,79,108],"\u2026,":[72],"N":[74,86,140,146],"i.i.d.r.v.":[76],"i":[80,109],"\u2032s":[81,110],"are":[82],"Unif,":[83],"then":[84,133],"\u2192":[87],"\u221e":[88],"differences":[96,143,150,180],"between":[97,190],"adjacent":[98],"order":[99,153],"statistics":[100,154],"converges":[101,181],"behavior.":[105,158,202],"instead":[107],"come":[111],"from":[112],"compactly":[114],"supported":[115],"with":[117],"uniformly":[118],"bounded":[119],"first":[120],"second":[122],"derivatives":[123],"second\u2010order":[126],"Taylor":[127],"series":[128],"expansion":[129],"at":[130],"each":[131],"point,":[132],"any":[139],"\u03b4":[141],"consecutive":[142],"all":[145,177],"\u2212":[147],"normalized":[149],"exhibit":[155],"We":[159],"derive":[160],"conditions":[161],"on":[162],"probability":[164],"density":[165],"which":[166],"determine":[167],"whether":[168],"or":[169,188],"not":[170],"unnormalized":[179],"law,":[184],"behavior,":[187],"oscillates":[189],"two,":[192],"show":[194],"Pareto":[197],"leads":[199],"oscillating":[201]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2016-06-24T00:00:00"}
