Lecture16IntroductiontoAsymptotics講座16介紹的漸近性Lecture16IntroductiontoAsymptotics講座16介紹的漸近性Lecture16IntroductiontoAsymptotics講座16介紹的漸近性Lecture16:

IntroductiontoAsymptotics(Chapter12.1–12.3)Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AgendaTheAsymptoticPerspective(Chapter12.1)AsymptoticUnbiasedness(Chapter12.2)Consistency(Chapter12.2)ProbabilityLimits(Chapter12.2)ConsistencyofOLS(Chapter12.3)3Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Lecture16IntroductiontoAsyLecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件Lecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件Lecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件Lecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件TheAsymptoticPerspective(cont.)Insteadoflookingatthepropertiesoftheestimatoraveragingoverallpossiblesamples,wewillstartlookingatpropertiesasthesamplesizegetsvery,verylarge.Whytheshift?Themathisgoingtobemuchmoreconvenient.Cross-samplepropertiesbecomeintractableaswerelaxthelast(andmostcrucial)Gauss–Markovassumption,thattheX

’sarefixedacrosssamples.6Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Large-sample(asymptotic)propertiesaremathematicallytractable.Wejusthavetohopethatestimatorsthatweproveworkwellwitha

near-infinitenumberofobservationswillalsoworkwellwiththefinitedatasetsweactuallyobserve.7Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)OneuseofMonteCarlotechniquesis

tostudycomputationallythesmallsamplepropertiesofestimatorsthathavebeenderivedasymptotically.Estimatorsdesignedforlargesamplesdon’ttendtoworkwellinsmallsamples,butareappropriatefor“reasonablylarge”samples.8Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Whataretheasymptoticproperties

ofinterest?Let’sstartwithsomeinformalexplanations.AsymptoticUnbiasedness:asthesamplesizegetsvery,verylarge,theestimatorbecomesunbiased(eventhoughtheremaybebiasesatsmallsamplesizes).9Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.Wecanseethispropertyinouroriginal

MonteCarlosimulationsofbg1-bg3.10Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coFigure12.1TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithNormallyDistributedDisturbances(1of2)11Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1TheDistributionsFigure12.1TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithNormallyDistributedDisturbances(2of2)12Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1TheDistributionsFigure12.1

TheDistributions

ofg1,g2,and

g3forSeveral

SampleSizeswithNormallyDistributedDisturbancesFigure12.1

TheDistributionTheAsymptoticPerspective(cont.)Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.WecanseethispropertyinouroriginalMonteCarlosimulationsofbg1-bg3.Whatifwerepeatthesimulations,

butdrawefromaskewed,non-

Normaldistribution?14Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(1of2)15Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(2of2)16Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2

TheDistributions

ofg1,g2,andg3

forSeveralSampleSizeswithSkewed,DiscreteDisturbancesFigure12.2

TheDistributionTheAsymptoticPerspectiveTheLawofLargeNumbers:

undersuitable(andeasilyattained)conditions,thesamplemeanisaconsistentestimatorofthecorrespondingpopulationmean.18Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspectiveTheTheAsymptoticPerspective(cont.)AsymptoticNormality:asthesamplesizegrowsvery,verylarge,theestimatorfollowstheNormaldistribution(eventhoughitmayfollowadifferentdistributioninsmallersamples).19Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Aswesawinthepreviousfigures,thedistributionofbg1,bg2,andbg3appearstobeNormalevenatsmallsamplesizeswhenMoresurprisingly,thedistributionofbg1,bg2,andbg3alsoappearstobeNormalatlargersamplesizesevenwheneisdistributedquitenon-Normally.20Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(1of2)21Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(2of2)22Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2

TheDistributions

ofg1,g2,andg3

forSeveralSampleSizeswithSkewed,DiscreteDisturbancesFigure12.2

TheDistributionTheAsymptoticPerspectiveTheCentralLimitTheorem:

undersuitable(andoftenreasonable)conditions,thedistributionofasample

meantendstobeapproximatelyNormallydistributedinlargesamples.TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.24Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspectiveTheTheAsymptoticPerspective(cont.)TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.Werarelyknowforsurethedistributionof

,soitisre-assuringtoknowourestimatorswilltypicallybeapproximatelyNormalin

largesamples,regardlessofitssmall-

sampledistribution.25Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)KeyTermssofar:AsymptoticAsymptoticUnbiasednessConsistencyTheLawofLargeNumbersAsymptoticNormalityTheCentralLimitTheorem26Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coAsymptoticUnbiasedness(Chapter12.2)Let’slookatsomeofthesepropertiesmoreclosely,startingwithasymptoticunbiasedness.Formanybiasedestimators,thebiasshrinkssmallerandsmallerasthesamplesizegrows.Asthesamplesizegrowsinfinitelylarge,thebiasshrinkstozero.Suchestimatorsareasymptoticallyunbiased.27Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(ChaptAsymptoticUnbiasedness(cont.)Forexample,wehavelearnedthatwemustmakea“degreesoffreedom”correctiontoourestimatedstandarderrors.28Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.AsymptoticUnbiasedness(cont.)Weknowthats2isanunbiasedestimateof

s2.Butwhatifweneglectedthedegreesoffreedomcorrection?29Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.AsymptoticUnbiasedness(cont.)30Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.AsymptoticUnbiasedness(cont.)31Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.AsymptoticUnbiasedness(cont.)n30.3333100.80001000.98001,0000.998010,0000.999832Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.CheckingUnderstandingWhichofthefollowingareasymptoticallyunbiasedestimatorsofbundertheGauss–Markovassumptions?33Copyright?2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstandingWhichofCheckingUnderstanding(cont.)34Copyright?2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)CheckingUnderstanding(cont.)35Copyright?2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)ConsistencyWewouldlovetobeguaranteedthatourestimatorwillbeexactlyright,oratleastvery,veryclosetoexactlyright.Certaintyistoomuchtoaskforinastochasticworld.However,ifourestimatorisconsistent,wecanbe“almostalwaysalmostright”invery,verylargesamples.36Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyWewouldlovetobeConsistency(cont.)Moreprecisely:anestimatorisconsistentif,providedwegetthesamplesizehighenough,theprobabilitycanbeascloseto1aswelikethatourestimateisasclosetobeingrightaswelike.Withfinitesamplesizes,thereisalwayssomeprobabilitythataconsistentestimatorisverywrong.Butasthesamplesizegrows,thatprobabilitybecomesvery,verysmall.37Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Consistency(cont.)MoreprecisFigure12.3TheCollapseofaConsistentEstimator’sDistributionasnGrows38Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.3TheCollapseofaConsistencyWhatdoesittaketogetconsistency?Oneoften-encounteredwayistohave

anasymptoticallyunbiasedestimatorwhosevarianceshrinkstozeroasthesample

sizegrows.Inlargesamples,theestimatoronaverage

isright.Inlargesamples,nosingleestimateislikelytobeveryfarfromtheestimator’saverage.39Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyWhatdoesittaketConsistency(cont.)Apathologicalexampleofconsistencywithoutasymptoticunbiasedness:40Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Consistency(cont.)ApathologiConsistency(cont.)41Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Consistency(cont.)41CopyrightProbabilityLimits(Chapter12.2)Howcanwedetermineifanestimatorisconsistentornot?Weneedanewmathematicaltool,theprobabilitylimit(orplim).Theplimisconceptuallymorecomplicatedthanexpectations,butinpracticeitiseasiertoworkwith.42Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(Chapter12ProbabilityLimits(cont.)Arandomvariablebconvergesinprobabilitytoaconstantvaluecif,asthesamplesizegrowsverylarge,theprobabilityapproaches1thatbtakesonavalueveryclosetoc.Wecallctheprobabilitylimitofb.plim(b)=c43Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)AraProbabilityLimits(cont.)44Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)44CoProbabilityLimits(cont.)45Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)45CoProbabilityLimits(cont.)46Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)46CoProbabilityLimits(cont.)47Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)47CoCheckingUnderstanding48Copyright?2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding48CopyriCheckingUnderstanding(cont.)49Copyright?2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)CheckingUnderstanding(cont.)50Copyright?2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)AlgebraofProbabilityLimits51Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AlgebraofProbabilityLimits5AlgebraofProbabilityLimits(cont.)52Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AlgebraofProbabilityLimitsConsistencyofOLS(Chapter12.3)Howdoweshowthatanestimator

isconsistent?Let’sstartwiththeGauss–Markovassumptions,andastraightlinethroughtheorigin.53Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(Chapter12ConsistencyofOLS(cont.)54Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)54CoConsistencyofOLS(cont.)First,notethatweknowbg4isunbiased,

soE(bg4)=b.Thus,plim(bg4)=b

iftheVar(bg4)

approaches0asngrows.Var(bg4)approaches0ifSXi2approaches

∞(ands2isfinite).55Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)FirsConsistencyofOLS(cont.)Var(bg4)approaches0ifSXi2

approaches∞(ands

2isfinite).Theseassumptionswilltypicallybemet,butwedoneedtoaugmentourGauss–Markovassumptionstoruleouttheunusualcases.56Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)Var(ConsistencyofOLS(cont.)Nowlet’saddanintercepttothemodel:57Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)NowConsistencyofOLS(cont.)Wecouldusethesamemethodweusedforbg4:showthatOLSisunbiasedandthatthevarianceshrinksto0as

ngrowslarger.Instead,we’regoingtousea

differentmethod.58Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)WecConsistencyofOLS(cont.)59Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)59CoConsistencyofOLS(cont.)60Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)60CoConsistencyofOLS(cont.)WenowneedtoaddanassumptiontotheGauss–MarkovDGP,toguaranteethatWeneedtoassumethat,asngrowslarge,61Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)WenConsistencyofOLS(cont.)62Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)62CoConsistencyofOLS(cont.)63Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)63CoConsistencyofOLS(cont.)64Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)64CoConsistencyofOLS(cont.)65Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)65CoConsistencyofOLS(cont.)66Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)66CoReview(cont.)Thislectureintroducesasetofnewcriteriaforjudgingestimators,basedonlarge-sample(asymptotic)properties.Insteadoflookingatthepropertiesoftheestimatoraveragingoverallpossiblesamples,welookatpropertiesasthesamplesizegetsvery,verylarge.67Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)ThislectureintReview(cont.)Whataretheasymptoticproperties

ofinterest?AsymptoticUnbiasedness:asthesamplesizegetsvery,verylarge,theestimatorbecomesunbiased(eventhoughtheremaybebiasesatsmallsamplesizes).68Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)WhataretheasyReview(cont.)Whataretheasymptoticproperties

ofinterest?Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.69Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)WhataretheasyReview(cont.)TheLawofLargeNumbers:undersuitable(andeasilyattained)conditions,thesamplemeanisaconsistentestimatorofthecorrespondingpopulationmean.70Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)TheLawofLargeReview(cont.)Whataretheasymptoticproperties

ofinterest?AsymptoticNormality:asthesamplesizegrowsvery,verylarge,theestimatorfollowstheNormaldistribution(eventhoughitmayfollowadifferentdistributioninsmallersamples).71Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)WhataretheasyReview(cont.)TheCentralLimitTheorem:

undersuitable(andreasonablyeasytoattain)conditions,thedistributionofasamplemeantendstobeapproximatelyNormallydistributedinlargesamples.TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.72Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)TheCentralLimiReview(cont.)KeyTermssofar:AsymptoticAsymptoticUnbiasednessConsistencyTheLawofLargeNumbersAsymptoticNormalityTheCentralLimitTheorem73Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)KeyTermssofarReview(cont.)Whatdoesittaketogetconsistency?Oneoften-encounteredwayistohave

anasymptoticallyunbiasedestimatorwhosevarianceshrinkstozeroasthesample

sizegrows.Inlargesamples,theestimatoronaverage

isright.Inlargesamples,nosingleestimateislikelytobeveryfarfromtheestimator’saverage.74Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)WhatdoesittakFigure12.3TheCollapseofaConsistentEstimator’sDistributionasnGrows75Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.3TheCollapseofaReviewHowcanwedetermineifanestimatorisconsistentornot?Weneedanewmathematicaltool,theprobabilitylimit(or

plim).76Copyright?2006PearsonAddison-Wesley.Allrightsreserved.ReviewHowcanwedetermineifReview(cont.)Arandomvariablebconvergesinprobabilitytoaconstantvaluecif,asthesamplesizegrowsverylarge,theprobabilityapproaches1thatbtakesonavalueveryclosetoc.Wecallctheprobabilitylimitofb.plim(b)=c77Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)ArandomvariablReview(cont.)78Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)78Copyright?20AlgebraofProbabilityLimits79Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AlgebraofProbabilityLimits7謝謝你的閱讀知識就是財富豐富你的人生謝謝你的閱讀知識就是財富謝謝！21、要知道對好事的稱頌過于夸大，也會招來人們的反感輕蔑和嫉妒。——培根

22、業精于勤，荒于嬉；行成于思，毀于隨。——韓愈

23、一切節省，歸根到底都歸結為時間的節省。——馬克思

24、意志命運往往背道而馳，決心到最后會全部推倒。——莎士比亞

25、學習是勞動，是充滿思想的勞動。——烏申斯基供婁浪頹藍辣襖駒靴鋸瀾互慌仲寫繹衰斡染圾明將呆則孰盆瘸砒腥悉漠塹脊髓灰質炎(講課2019)脊髓灰質炎(講課2019)謝謝！21、要知道對好事的稱頌過于夸大，也會招來人們的反感輕Lecture16IntroductiontoAsymptotics講座16介紹的漸近性Lecture16IntroductiontoAsymptotics講座16介紹的漸近性Lecture16IntroductiontoAsymptotics講座16介紹的漸近性Lecture16:

IntroductiontoAsymptotics(Chapter12.1–12.3)Copyright?2006PearsonAddison-Wesley.Allrightsreserved.AgendaTheAsymptoticPerspective(Chapter12.1)AsymptoticUnbiasedness(Chapter12.2)Consistency(Chapter12.2)ProbabilityLimits(Chapter12.2)ConsistencyofOLS(Chapter12.3)3Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Lecture16IntroductiontoAsyLecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件Lecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件Lecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件Lecture-16-Introduction-to-Asymptotics講座16介紹的漸近性共8課件TheAsymptoticPerspective(cont.)Insteadoflookingatthepropertiesoftheestimatoraveragingoverallpossiblesamples,wewillstartlookingatpropertiesasthesamplesizegetsvery,verylarge.Whytheshift?Themathisgoingtobemuchmoreconvenient.Cross-samplepropertiesbecomeintractableaswerelaxthelast(andmostcrucial)Gauss–Markovassumption,thattheX

’sarefixedacrosssamples.87Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Large-sample(asymptotic)propertiesaremathematicallytractable.Wejusthavetohopethatestimatorsthatweproveworkwellwitha

near-infinitenumberofobservationswillalsoworkwellwiththefinitedatasetsweactuallyobserve.88Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)OneuseofMonteCarlotechniquesis

tostudycomputationallythesmallsamplepropertiesofestimatorsthathavebeenderivedasymptotically.Estimatorsdesignedforlargesamplesdon’ttendtoworkwellinsmallsamples,butareappropriatefor“reasonablylarge”samples.89Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Whataretheasymptoticproperties

ofinterest?Let’sstartwithsomeinformalexplanations.AsymptoticUnbiasedness:asthesamplesizegetsvery,verylarge,theestimatorbecomesunbiased(eventhoughtheremaybebiasesatsmallsamplesizes).90Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.Wecanseethispropertyinouroriginal

MonteCarlosimulationsofbg1-bg3.91Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coFigure12.1TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithNormallyDistributedDisturbances(1of2)92Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1TheDistributionsFigure12.1TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithNormallyDistributedDisturbances(2of2)93Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1TheDistributionsFigure12.1

TheDistributions

ofg1,g2,and

g3forSeveral

SampleSizeswithNormallyDistributedDisturbancesFigure12.1

TheDistributionTheAsymptoticPerspective(cont.)Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.WecanseethispropertyinouroriginalMonteCarlosimulationsofbg1-bg3.Whatifwerepeatthesimulations,

butdrawefromaskewed,non-

Normaldistribution?95Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(1of2)96Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(2of2)97Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2

TheDistributions

ofg1,g2,andg3

forSeveralSampleSizeswithSkewed,DiscreteDisturbancesFigure12.2

TheDistributionTheAsymptoticPerspectiveTheLawofLargeNumbers:

undersuitable(andeasilyattained)conditions,thesamplemeanisaconsistentestimatorofthecorrespondingpopulationmean.99Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspectiveTheTheAsymptoticPerspective(cont.)AsymptoticNormality:asthesamplesizegrowsvery,verylarge,theestimatorfollowstheNormaldistribution(eventhoughitmayfollowadifferentdistributioninsmallersamples).100Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)Aswesawinthepreviousfigures,thedistributionofbg1,bg2,andbg3appearstobeNormalevenatsmallsamplesizeswhenMoresurprisingly,thedistributionofbg1,bg2,andbg3alsoappearstobeNormalatlargersamplesizesevenwheneisdistributedquitenon-Normally.101Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(1of2)102Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(2of2)103Copyright?2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsFigure12.2

TheDistributions

ofg1,g2,andg3

forSeveralSampleSizeswithSkewed,DiscreteDisturbancesFigure12.2

TheDistributionTheAsymptoticPerspectiveTheCentralLimitTheorem:

undersuitable(andoftenreasonable)conditions,thedistributionofasample

meantendstobeapproximatelyNormallydistributedinlargesamples.TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.105Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspectiveTheTheAsymptoticPerspective(cont.)TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.Werarelyknowforsurethedistributionof

,soitisre-assuringtoknowourestimatorswilltypicallybeapproximatelyNormalin

largesamples,regardlessofitssmall-

sampledistribution.106Copyright?2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(coTheAsymptoticPerspective(cont.)KeyTermssofar:AsymptoticAsymptoticUnbiasednessConsistencyTheLawofLargeNumbersAsymptoticNormalityTheCentralLimitTh