数理统计学导论参考答案(第6-9章)R.V霍格

SolutiontoExercisesChapter6IntroductiontoStatisticalInferenceSection6.1PointEstimation6.1.LetnXXX,,,21representarandomsamplefromeachofthedistributionshavingthefollowingprobabilitydensityfunctions：(a）0,,2,1,0,!/),(xxexfx,zeroelsewhere,where.1)0,0(f(b)0,10,),(1xxxf,zeroelsewhere.(c)0,0,1),(/xexfx,zeroelsewhere.Ineachcasefindthem.l.e.ˆof.Solution(a)Thelikelihoodfunctionofthesampleis!!!/();(21nnxxxxexLiHere!lnln);(lniixnxxLSowehave.0)(lnnxdLdiwhosesolutionforisxwhichisthedesiredm.l.e.oftheunknownparameter.(b)Thelikelihoodfunctionofthesampleis121)();(nnxxxxLHere)ln)(1(ln);(lnixnxLSowehave.0ln)(lnixndLdwhosesolutionforisixnln/whichisthedesiredm.l.e.oftheunknownparameter.(c)Thelikelihoodfunctionofthesampleis/1);(ixnexLHere/ln);(lnixnxLSowehave.0/)(ln2ixndLdwhosesolutionforisxwhichisthedesiredm.l.e.oftheunknownparameter.6.2.LetnXXX,,,21bei.i.d.,eachwiththedistributionhavingp.d.f.211/)(2210,,,)/1(),;(21xexfx,zeroelsewhere.Findthem.l.e.of1and2.SolutionGiven2,itiseasilyverifythatthefirstorderstatisticcanmaximizethelikelihoodfunction,sothem.l.e.of1isthefirstorderstatistic1Y.Thelikelihoodfunctionofthesampleis211/)(2210,,,)/1(),;(21xexLixn.21221/)(ln),;(lnixnxLWeobservethatwemaymaximizebydifferentiation.Wehave0)(ln22122ixnLwhosesolutionisnYXi/)(1whichisthem.l.e.oftheunknownparameter2.6.3.LetnYYY21betheorderstatisticsofarandomsamplefromadistributionwithp.d.f.,,2121,1);(xxfzeroelsewhere.Showthateverystatistic),,,(21nXXXusuchthat21),,,(2121nnnYXXXuYisam.l.e.of.Inparticular,6/)142(and2/)(,6/)124(111nnnYYYYYYarethreesuchstatistics.Thustheuniquenessisnotingeneralapropertyofam.l.e.SolutionAccordingtothedefinitionoftheorderstatistic,wehave21)2121nYXY.Fromtheinequality,weobtain21211YYnwhichmeansthatanystatistic),,,(21nXXXusuchthat21),,,(21121YXXXuYnnisam.l.e.of.Particularly,thestatistics21and211YYnarebothm.l.e.of.Furthermore,anyweightyaverageofthetwostatisticsism.l.e.Sincethestatisticscanbeformulatedas).21(62)21(646/)142(),21(21)21(212/)(),21(64)21(626/)124(111111YYYYYYYYYYYYnnnnnnSo6/)142(and2/)(,6/)124(111nnnYYYYYYarethreem.l.e.of.6.4.Let321and,XXXhavethemultinomialdistributioninwhich4,25kn,andtheunknownprobabilitiesare321and,,respectively.Herewecan,forconvenience,let321432141and25XXXX.Iftheobservedvaluesoftherandomvariablesare,7and,11,4321xxxfindthem.l.e.of321and,.SolutionItiseasilytounderstandthat3,2,1),,25(~ibXiiSothem.l.e.oftheunknownparametersis321,,XXX,respectively.Thusthem.l.e.of321and,is257,2511,254,respectively.6.5.TheParetodistributionisfrequentlyusedasamodelinstudyofincomesandhasthedistributionfunction0and0whereelsewhere,zero,,)/(1),;(2111212xxxFIfnXXX,,,21isarandomsamplefromthisdistribution,findthem.l.e.of21and.SolutionThep.d.f.ofthepopulationisxxxf111221,),;(22.Obviously,them.l.e.of1isthefirstorderstatistic1Y.Thelikelihoodfunctionofthesampleis0lnlnln,),;(1221121112212222innnxnnLxxxxLThusweobtainthem.l.e.of212lnlnˆYnXni.6.6.LetnYbeastatisticsuchthat0limand)(lim2nnYnnYE.ProvethatnYisconsistentestimatorof.ProofSince2222)]([]))()([(])[(nYnnnnYEYEYEYEYEn,So,inaccordancewithChebyshev’sinequality,wehavenYEYEYnYnnnas,0)]([])[()|Pr(|22222forevery0.Thusaccordingtothedefinitionofconsistentestimator,wecompletetheproof.6.7.ForeachofthedistributionsinExercise6.1,findanestimatorofbythemethodofmomentsandshowthatitisconsistent.Solution(1)ItisobviousthatthepopulationisPoissondistributionwithparameter.So)(XE.LetX.WegettheestimatorofbythemethodofmomentsisthesamplemeanX.nXVXE)(,)(.Forany0,wehavennXVXas,0)()|Pr(|22ThusthesamplemeanXisaconsistentestimatorofthepopulationmean.(2)Thepopulationmeanis.1d)(10xxXEInaccordancewiththeideaofmethodofmoments,letX1WehaveXX1ˆwhichisthemomentestimatorof.(3)Infact,thepopulationisGammadistributionwithparameters1and.So)(XE.ThustheestimatorofbymethodofmomentsisthesamplemeanX.ItiseasilytoverifythatthesampleXconvergesinprobabilitytothepopulationmean,soXisaconsistentestimatorof.6.8.Ifarandomsampleofsizenistakenfromadistributionhavingp.d.f.,0,/2);(2xxxfzeroelsewhere,find(a)Them.l.e.ˆfor.(b)Theconstantcsothat)ˆ(cE.(c)Them.l.e.forthemedianofthedistribution.Solution(a)ThelikelihoodfunctionofthesampleisnnnxxxxL221/2);(.ItisobviousthatthethnorderstatisticnYcanmaximizethelikelihoodfunction,sothem.l.e.ˆforisthethnorderstatisticnY.b)(b)sincethep.d.f.ofnYisnnnnnynyyf0,/2)(212,thus122)(nnYEnSonnc212.(c)Since,2/,//2212202mmdxxmInaccordancewiththeinvariantpropertyofm.l.e.wehave2/ˆnYm.6.9.LetnXXX,,,21bei.i.d.,eachwithadistributionwithp.d.f.xexfx0,)/1();(/,zeroelsewhere.Findthem.l.e.of)2Pr(X.SolutionItisnotdifficulttofindthatthem.l.e.ofisthesamplemeanX.