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1、7.EstimationProblems7.1PointEstimate点估计1.conceptEstimationisoneofthemostimportantproblemsinstatistics.Why----thedistributiondependsontheparameter.PoissondistributionNormaldistributionProblemsofestimationinclude(1)determiningparametersofapopulationfromtherandomsamples,estimatingthemeanorvar
2、ianceofapopulationbysampling,(2)howtojudgeifsuchestimationisa“good”one.PointEstimation-----usethevalueofastatistictoestimateaparameterDefinition7.1.1Supposeisaparameterofapopulation,isarandomsamplefromthispopulation,andisastatisticthatisafunctionof.Now,totheobservedvalue,ifweuseasanestimat
3、edvalueof,theniscalledapointestimatorofandisreferredasapointestimateof.Thepointestimatorisalsooftenwrittenas.Forexample,ifweusetoestimatetheparameterofaPoissonpopulation,isthentheestimatorofandisapointestimateoftheparameter.Pointestimationisnotunique.Forexample,weoftenuseor-----mean.2.Judg
4、e(1)Unbiased无偏(2)asymptoticallyunbiased一致(3)minimumvarianceunbiasedestimator.最有效无偏估计Definition7.1.2.Supposeisanestimatorofaparameter.ThenisunbiasedifandonlyifExample7.1.1IfisthesamplemeanofarandomsampleofsizefromaPoissonpopulation,showthatisanunbiasedestimatorof.Proof.SinceforaPoissonwekno
5、wthatandbyTheorem6.3.1,weconcludethatwhichshowsthatisanunbiasedestimatorof.□Example7.1.2Letbethesamplemeanofrandomsampleofsizefromapopulationwithprobabilitydensitygivenby.Showthatisabiasedestimatorof.Proof.Supposeisthemeanofthepopulation,byTheorem6.4.1（P100）,wehavethat.Therefore,isabiasede
6、stimatorof.均值的点估计Firstofall,letusdiscussthesituationofusingpointestimations.isusuallythemostcommonchoiceofthepointestimatorofthemean.ByTheorem6.4.1,wecaneasilyobtainthefollowingtheorem.Letbethemeanofarandomsampleofsizefromarandomvariablewhichhasmean,thenisanunbiasedestimatorof.Definition7.
7、1.2isanasymptoticallyunbiasedestimatorofif.Example7.1.3Letbethevarianceofarandomsampleofsizefromarandomvariablewhichhasafinitevariance.Showthatisanasymptoticallyunbiasedestimatorof.Proof.ByTheorem6.3.1,.Sinceand,wehaveHere,.Thus,wegetthat.Wecanseethatisabiased
