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1、MULTIVARIATELIOUVILLEDISTRIBUTIONSby12RameshwarD.GuptaDonaldSt.P.Richards,3DivisionofMathematics.DepartmentofStatisticsComputerScienceandEngineeringUniversityofNorthCarolinaUniversityofNewBrunswickChapelHill.N.C.27514St.John.NewBrunswick.Canada1partiallysupportedbytheNSERCresearchgrantA-4850.2par
2、tiallysupportedbytheNationalScienceFoundationongrantMCS-8403381.3PortionsofthisresearchwerecompletedduringarecentleaveofabsenceattheUniversityofWyoming.AMS1980SubjectClassification:Primary62H10,Secondary62E15,28A01.KeyWordsandPhrases:Dirichletdistribution,Dirichletintegral,regression,totallyposit
3、ive,reverserule,stochasticrepresentation,characterization,multivariateDirichletdistribution,fractionalintegral.ABSTRACTArandomvector(Xl'•..,X),withpositivecomponents,issaidntohaveaLiouvilledistributionifitsjointprobabilitydensityaCIan-lfunctionisoftheformf(xl+...+x)xl•..xwiththeanniallpositive.Ex
4、amplesofthesearetheDirichletandinvertedDirichletdistribution.Inthispaper,acomprehensivetreatmentoftheLiouvilledistributionsisprovided.Theresultsdevelopedpertaintostochasticrepresentations,transformationproperties,completeneutrality,marginalandconditionaldistributions,regressionfunctions,totalposi
5、tivityandreverseruleproperties.Further,thesetopicsareutilizedinvariouscharacterizationsoftheDirichletandinvertedDirichletdistributions.MatrixanalogsoftheLiouvilledistributionsarealsodefined,andmanyoftheresultsobtainedinthevectorsettingareextendedappropriately.1.INTRODUCTIONDuringrecentyears,incre
6、asingattentionhasbeenpaidtofamiliesofprobabilitydistributionswhicharedefinedthroughvariousfunctionalformassumptions.JohnsonandKotz[12;p.294ff]reviewanumberofarticleswhichclassifydistributionsthroughfunctionalformrequirementsondensityfunctions.Evenmorecontemporaryresultsbeginwithfunctionalformrest
7、rictionsoncharacteristicfunctions,ratherthanonthedensities.WereferthereadertoCambanisetal[1],[2]andRichards[25]forexamplesofthelatterapproach.Therearetwoimportantconsequencesofthesestudies.First,theydevelopunifiedanaly
