Density of the ratio of two normal random variable

1DENSITYOFTHERATIOOFTWONORMALRANDOMVARIABLESbyT.Pham-Gia*andN.TurkkanUniversitedeMonctonandE.Marchand,UniversityofNewBrunswickCANADAAbstract:Wegivetheexactclosedformexpressionofthedensityof12/XX,where1Xand2Xarenormalrandomvariables,intermsofHermiteorConfluentHypergeometricfunctions.Allcaseswillbeconsidered:standardizedandnonstandardizedvariables,independentorcorrelatedvariables.Severalnewapplicationsarepresented,andrelationshipswithmixturesofnormaldistributionsaregiven.KeywordsandPhrases:Normal,Bivariatenormal,Ratio,Hermitefunction,Kummerconfluenthypergeometricfunction,Integralrepresentation,FinitesamplingAMSClassification:62E15,62N05.1.INTRODUCTIONThedensityofof/WXY=,whereXandYarenormalrandomvariables,hasattractedtheinterestofseveralresearchers,asearlyas1930,sinceitwasencounteredinsomebasicproblemsinStatistics.AlthoughthecaseswherebothXandYarestandardnormal,orstandardbivariatenormal,arefairlysimple,generalcasesaremuchmorecomplex.Geary(1930)wasthefirsttoinvestigatethisquestionandFieller(1932)presentedanotherapproachtoevaluatethisdensity.Inthesixties,twoimportantpapers(Marsaglia(1965)andHinkley(1969))bothaddressedthisconcern,withdifferentviewpoints,however.Recently,demandforthisexpressionresurfaceinanumberofimportantapplications,andanactiveexchangeontheInternetMathforum(Startz(1997),Ward(1997)andMarsaglia(2001)),hasrekindledtheneedforaconvenientexpressionofthisdistribution.Springer(1984,p.139),usingMellinTransformmethods,haspresentedamethodtoobtainthisdensity,intermsofaninfiniteseries,buttheinversionoftheMellintransforminthecomplexplanenaturallyrequiressomeadvancedcomputation,thatisnotalwayseasytohandle.Althoughthismethoddoesguaranteetheresults,muchnumericalanalysishastobeperformed,andtheresultisnotaclosedformexpression.Inthisarticlewewillusespecialmathematicalfunctions,theHermitefunction,whichisageneralizationoftheHermitepolynomials,andKummer’sconfluenthypergeometricfunction,togiveaconvenientclosedformexpressionforthedensityof/XY.Thisexpressioncanalsobeobtainedusingacompletelydifferentapproach,basedonconditionalexpectation.Someparticularcaseshave,however,simplerexpressions,expressedascommonfunctions.Insection2,wewillfirstrecalltheHermitefunction()Hzν,andgiveitsbasicproperties,anditsintegralrepresentationwhenνisanegativerealnumber.Insections3and4,thedensityofWisgivenfordifferent,andexhaustingcases.Section5discussestheinterestingshapesthatthedensityofWcantake,whilesection5presentssomenumericalapplications,with__________________________________________________________________________*)ResearchpartiallysupportedbyNSERCgrantA9249(Canada).2relateddiscussions.Finally,section6givesanotherlookattheproblem,fromacumulativedistributionviewpoint,andputsitinitswidercontextofcontinuousanddiscretemixtureofnormaldistributions.2.THEHERMITEFUNCTIONandTHEPOWER-QUADRATICEXPONENTIALFAMILYAlthoughHermitepolynomialsarewellutilizedinStatistics,forinstanceintheGram-Charlierexpansionofadensity,theHermitefunctionhasonlytimidencounterswithdistributiontheory(Pham-Gia(1992)).Theuseofspecialfunctions,alreadyverywidespreadinMathematicalPhysics,isgaininggroundinStatistics,wheretheyprovidepowerfultoolstocomplementtheclassicalcommonfunctions.Dickey(1983)haschampionedsuchause,alreadyafewdecadesago.TheHermitefunctionwithparameterν,()Hzν,canbederivedfromtheparaboliccylinderfunctionDνbytherelation/22()2exp(/2)(2),HzzDzννν=whereDνitselfisrelatedtothethν−derivativeofthefunction2exp(/2)z−bytherelation:22/2()(1)exp(/4)()zdDzzedzνννν−=−,1.νForanyvalueofν,Hνcanbedirectlydefinedbytheinfiniteseries(Lebedev(1972,p.289)):01(1)(()/2)()(2)2()!nmmmHzzmννν∞=−Γ−=Γ−∑,(1)withtheGammafunctionfornegativevaluesobtainedbyrepeatedlyapplyingtherelation:1()(/2)((1)/2)/[2],0ννννπν+Γ−=Γ−Γ−and(1/2)(3/2)πΓ−Γ=−.Whenνisapositiveinteger,wehavethecorrespondingHermitepolynomial,andwith0ν,()Hzνhasanintegralrepresentationoftheform:22(1)01()()ttzHzetdtννν∞−−−+=Γ−∫,(2)whichshowsthat()Hxνisapositivefunctiononthewholerealline.TheHermitefunction2()Hz−isofparticularinterestinthisarticle.Wehave:2220()ttzHztedt∞−−−=∫,(3)with2(0)/2Hπ−=,andfromthegeneralrelation21(1)22(1)/202()()(/2)tHzzettzdtννννν∞+−−+−=+Γ−∫,0ν(Lebedev(1972,p.297)wehave:22223/20()2()tzetHzdttz∞−−=+∫.AnimportantidentityrelatestheHermitefunctiontoKummer’sclassicalconfluenthypergeometricfunctionoffirstkind,11F,definedby:3110(,)(,;).(,)!kkkzFzkkααγγ∞==∑,0,1,2,..γ≠−−,,withtheascendingfactorial,orPochhamercoefficients,(,)(1)...(1)()/(),kkkαααααα=++−=Γ+Γand(,0)1α=.Itis:22111112()2[.(/2;1/2;).((1)/2;3/2;)1()()22zHzFzFzννπνννν=−−−−ΓΓ−(4)2.2AspresentedinPham-Gia(1994),thepower-quadraticexponentialfamilyofdistributionshasthepropertythatitshazardratescanbeorderedundercertainconditions.FurtherpropertiesontheratiosoftwomembersofthisfamilyarepresentedinPham-GiaandTurkkan(2004).DEFINITION:Thefamily(;,)PQEγδεofdistributionsconsistsofpositivecontinuousrandomvariableswithdensitiesoftheform:2(;;,)(;,)exp[()],ftCtttγγδεγδεδε=−+wherethedomainoftheparametersisasfollows:(;,){1;,0}γδεγδε∈−−∞∞U{1;0,0}γδε−=.Wewrite~(;,)XPQEγδε,andhavethefo