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1、Chapter7PointandIntervalEstimationInastronomy,noexperimentshouldbebelieveduntilconfirmedbytheory.SirArthurEddington,Britishastronomer(1882-1944)7.1Orientation“Youdon’thavetoeatawholeoxtoknowthemeatistough,”sowroteSamuelJohnson(1709-1784),Englishcriticandlexicographer.Itisanamusingreminde
2、rthataprimarygoalofastatisticalanalysisistomakeinferencesaboutapopulationbasedonasampledrawnfromthatpopulation.Awidelyusedmodelinfinancialengineering,forexample,assumesthatthestockpriceshavealognormaldistribution;moreprecisely,itisassumedthatln(S(t)/S)isnormallydistributedwithmeantµandva
3、riancetσ2(Proposition5.6).0Infinancialengineeringσiscalledthevolatilityandplaysanimportantroleinestimatingthevalueatrisk(VaR),tobediscussedinSection7.3.4,aswellasintheBlack-Scholesoptionpricingformuladiscussedearlier(Theorem5.9).Usinganincorrectvalueforthevolatilitycanleadtounacceptablyl
4、argelossesinaportfolio.Consequently,itisaquestionofnosmallimportancetodeterminetheaccuracyandreliabilityofaparameterestimate.Inpractice,theseparameterscanonlybeestimatedfromhistoricalstockmarketdataofthesortlistedinTables1.19and1.20.Itsometimeshappens,however,thatapopulationparameter,su
5、chasthepopulationmean,canbeestimatedbytwo(orevenmore!)differentmethods.Forexample,onecanuseeitherthesamplemeanXorsamplemedianXtoestimatethemeanofanormalpopulation.Whichoneisbetter?Andwhatcriteriadoweusetocomparethem?Thesearesomeofthequestionswewilldiscussinthischapter.OrganizationofChap
6、ter1.Section7.2:EstimatingPopulationParameters:MethodsandExamples2.Section7.3:ConfidenceIntervalsfortheMeanandVariance3.Section7.3.4:ValueatRisk(VaR):AnApplicationofConfidenceIntervalstoRiskManagement4.Section7.4:PointandIntervalEstimationfortheDifferenceofTwoMeans5.Section7.5:PointandInte
7、rvalEstimationforaPopulationProportion6.Section7.6:SomeMethodsofEstimation,includingthemethodofmomentsandthemaximumlikelihoodmethod*7.Section7.7:ChapterSummary291292IntroductiontoProbabilityandStatisticsforScience,Engineering,andFinance7.2EstimatingPopulationParameters:Methodsa
