Lectures.on.Statistics_Ash

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1、1LecturesOnStatisticsRobertB.AshPrefaceThesenotesarebasedonacoursethatIgaveatUIUCin1996andagainin1997.Nopriorknowledgeofstatisticsisassumed.Astandardfirstcourseinprobabilityisapre-requisite,butthefirst8lecturesreviewresultsfrombasicprobabilitythatareimportantinstatistics.Someexposur

2、etomatrixalgebraisneededtocopewiththemultivariatenormaldistributioninLecture21,andthereisalinearalgebrareviewinLecture19.Herearethelecturetitles:1.TransformationofRandomVariables2.Jacobians3.Moment-generatingfunctions4.Samplingfromanormalpopulation5.TheTandFdistributions6.Ordersta

3、tistics7.Theweaklawoflargenumbers8.Thecentrallimittheorem9.Estimation10.Confidenceintervals11.Moreconfidenceintervals12.Hypothesistesting13.Chisquaretests14.Sufficientstatistics15.Rao-Blackwelltheorem16.Lehmann-Scheff´etheorem17.Completesufficientstatisticsfortheexponentialclass18.Bayese

4、stimates19.Linearalgebrareview20.Correlation21.Themultivariatenormaldistribution22.Thebivariatenormaldistribution23.Cram´er-Raoinequality24.Nonparametricstatistics25.TheWilcoxontest
ccopyright2007byRobertB.Ash.Paperorelectroniccopiesforpersonalusemaybemadefreelywithoutexplicitperm

5、issionoftheauthor.Allotherrightsarereserved.1Lecture1.TransformationofRandomVariablesSupposewearegivenarandomvariableXwithdensityfX(x).WeapplyafunctiongtoproducearandomvariableY=g(X).WecanthinkofXastheinputtoablackbox,andYtheoutput.WewishtofindthedensityordistributionfunctionofY.We

6、illustratethetechniquefortheexampleinFigure1.1.2YY=Xf(x)X1/2y1-e-x2-Sqrt[y]Sqrt[y]X-1x-axisFigure1.1ThedistributionfunctionmethodfindsFYdirectly,andthenfYbydifferentiation.√√WehaveFY(y)=0fory<0.Ify≥0,thenP{Y≤y}=P{−y≤x≤y}.Case1.0≤y≤1(Figure1.2).Then√1√y11√1√F(y)=y+e−xdx=y+(1−e−y).Y2

7、0222f(x)X1/2-1Sqrt[y]x-axis-Sqrt[y]Figure1.2Case2.y>1(Figure1.3).Then√1y111√F(y)=+e−xdx=+(1−e−y).Y20222ThedensityofYis0fory<0and2f(x)X1/2'-11Sqrt[y]x-axis-Sqrt[y]Figure1.31√f(y)=(1+e−y),01.Y√4ySeeFigure1.4forasketchoffYandFY.(YoucantakefY(y)tobeanythingyouli

8、keaty=1because{Y=1}hasprobability