On the limit behaviour of the joint distribution function of order statistics
Identifieur interne : 000C02 ( Istex/Corpus ); précédent : 000C01; suivant : 000C03On the limit behaviour of the joint distribution function of order statistics
Auteurs : H. Finner ; M. RotersSource :
- Annals of the Institute of Statistical Mathematics [ 0020-3157 ] ; 1994-06-01.
Abstract
Abstract: Fork ∈ ℕ0 fixed we consider the joint distribution functionF n k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF n k (x 1,⋯,x n-k) in terms of the limit behaviour ofn(1-F(x n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution functionF.
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DOI: 10.1007/BF01720590
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Finner</name><affiliation><mods:affiliation>FB IV-Mathematik/Statistik, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Roters, M" sort="Roters, M" uniqKey="Roters M" first="M." last="Roters">M. 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The main result of the paper is a complete characterization of the limit behaviour ofF n k (x 1,⋯,x n-k) in terms of the limit behaviour ofn(1-F(x n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution functionF.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>H. Finner</name><affiliations><json:string>FB IV-Mathematik/Statistik, Universität Trier, D-54286, Trier, Germany</json:string></affiliations></json:item><json:item><name>M. 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This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. 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Amer. Statist. Assoc</title><publicationDate>1992</publicationDate></host><title>A step-up multiple test procedure</title><publicationDate>1992</publicationDate></json:item><json:item><host><author><json:item><name>H Finner</name></json:item><json:item><name>A,J Hayter</name></json:item><json:item><name>M Roters</name></json:item></author><title>On a conjecture concerning the joint distribution function of order statistics with special reference to stepwise multiple test procedures</title><publicationDate>1993</publicationDate></host></json:item><json:item><author><json:item><name>M,R Leadbetter</name></json:item><json:item><name>G Lindgren</name></json:item><json:item><name>H Rootz@n</name></json:item></author><host><author></author><title>Extremes and Related Properties of Random Sequences and Processes</title><publicationDate>1983</publicationDate></host><publicationDate>1983</publicationDate></json:item><json:item><host><author><json:item><name>R,D 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The main result of the paper is a complete characterization of the limit behaviour ofF n k (x 1,⋯,x n-k) in terms of the limit behaviour ofn(1-F(x n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. 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function of order statistics</ArticleTitle><ArticleCategory>Distribution</ArticleCategory><ArticleFirstPage>343</ArticleFirstPage><ArticleLastPage>349</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>5</Month><Day>3</Day></RegistrationDate><Received><Year>1992</Year><Month>12</Month><Day>11</Day></Received><Revised><Year>1993</Year><Month>6</Month><Day>11</Day></Revised></ArticleHistory><ArticleCopyright><CopyrightHolderName>The Institute of Statistical Mathematics</CopyrightHolderName><CopyrightYear>1994</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants><ArticleContext><JournalID>10463</JournalID><VolumeIDStart>46</VolumeIDStart><VolumeIDEnd>46</VolumeIDEnd><IssueIDStart>2</IssueIDStart><IssueIDEnd>2</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>H.</GivenName><FamilyName>Finner</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>M.</GivenName><FamilyName>Roters</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>FB IV-Mathematik/Statistik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><Postcode>D-54286</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>For<Emphasis Type="Italic">k</Emphasis> ∈ ℕ<Subscript>0</Subscript> fixed we consider the joint distribution function<Emphasis Type="Italic">F</Emphasis><Stack><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Superscript>k</Superscript></Stack> of the<Emphasis Type="Italic">n-k</Emphasis> smallest order statistics of<Emphasis Type="Italic">n</Emphasis> real-valued independent, identically distributed random variables with arbitrary cumulative distribution function<Emphasis Type="Italic">F</Emphasis>. The main result of the paper is a complete characterization of the limit behaviour of<Emphasis Type="Italic">F</Emphasis><Stack><Subscript>n</Subscript><Superscript>k</Superscript></Stack> (<Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript>,⋯,<Emphasis Type="Italic">x</Emphasis><Subscript>n-k</Subscript>) in terms of the limit behaviour of<Emphasis Type="Italic">n</Emphasis>(1-<Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis><Subscript>n</Subscript>)) if<Emphasis Type="Italic">n</Emphasis> tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (<Emphasis Type="Italic">k</Emphasis>+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution function<Emphasis Type="Italic">F</Emphasis>.</Para></Abstract><KeywordGroup Language="En"><Heading>Key words and phrases</Heading><Keyword>Extreme order statistic</Keyword><Keyword>extreme value distribution</Keyword><Keyword>joint distribution function of order statistics</Keyword><Keyword>multiple comparisons</Keyword><Keyword>Poisson distribution</Keyword><Keyword>uniform distribution</Keyword></KeywordGroup></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>On the limit behaviour of the joint distribution function of order statistics</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>On the limit behaviour of the joint distribution function of order statistics</title></titleInfo><name type="personal"><namePart type="given">H.</namePart><namePart type="family">Finner</namePart><affiliation>FB IV-Mathematik/Statistik, Universität Trier, D-54286, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">M.</namePart><namePart type="family">Roters</namePart><affiliation>FB IV-Mathematik/Statistik, Universität Trier, D-54286, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper"></genre><originInfo><publisher>Kluwer Academic Publishers</publisher><place><placeTerm type="text">Dordrecht</placeTerm></place><dateCreated encoding="w3cdtf">1992-12-11</dateCreated><dateIssued encoding="w3cdtf">1994-06-01</dateIssued><copyrightDate encoding="w3cdtf">1994</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: Fork ∈ ℕ0 fixed we consider the joint distribution functionF n k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF n k (x 1,⋯,x n-k) in terms of the limit behaviour ofn(1-F(x n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. 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