Review of Inferential Statistics January 21, 20041Political Science 552Review of Inferential StatisticsAlgebra of Summation8765432183XXXXXXXXXii+++++++=∑=223222112...NNiiXXXXX ++++=∑=NNNiiiYXYXYXYXYX ++++=∑=...3322111Sum of Square:Sum of Product:Summation-Continued()()( )( )( )()()∑∑∑===+=+++++++++=+++++++++=++++++++=++++++++=+NiiNiiNNNNNNNNNiiiYXYYYYXXXXYYYYXXXXYXYXYXYXYXYXYXYXYX113213213213213322113322111..................Sum of Sum:Review of Inferential Statistics January 21, 20042Summation-Continued II()∑∑===++++=++++=NiiNNNiiXkXXXXkkXkXkXkXkX13213211......()()∑∑∑∑∑∑∑∑========++=++=++=+NiiNiiiNiiNiiNiiiNiiNiiiiiNiiiYYXXYYXXYYXXYX121121211212212222Sum of Constant times Variable:Sum of Squared Sum:Summation-Continued III()∑∑=−==+++≠++++++++++=++++=NiiNNNNNNNiiXXXXXXXXXXXXXXXXXXXXXX12222211321312122221232121...2..22...22......rcrcjriijXXXXXXX +++++++=∑∑==.....22121211111Square of Sum:Double summation:Binomial Distribution()()()sNssNssNsNsNsp−−−−=−=ππππ1!!!1)(Review of Inferential Statistics January 21, 20043Notation Probability)()(ssYYPYf ==Joint ProbabilityConditional ProbabilityStatistical Independence),(21YYf);(21YYf)()(),(2121YfYfYYf ×=Expected Values{}∑==XssXfXXEµ)({} ( )∫+∞∞−= dXXXfXEFunctions and Variance)( XgY =()2)( kXXgY −=={} {}(){}22XEXEX −=σVARIANCE OPERATOR:{}X2σReview of Inferential Statistics January 21, 20044Algebra of Expectation}{}{ YEkYkE +=+}{}{ YEkYkE ×=× }{}{}{ YEXEYXE ×=×Sum of constant and random variable:Product of constant and random variableProduct of two independent random variables:kkE =}{ConstantExpectation-continued{}∑∑===kiikiiYEYE11Sum of two or more random variables:{}∑∑===kiiikiiiYEaYaE11Linear combination of two or more random variablesExpectation: Variances{}∑∑===kiiikiiiYaYa12212σσVariance of linear combination of two or more independent random variables:Covariance defined:{} {}(){}(){}YEYXEXEYX −−=,σCovariance of two linear combinations of the same set of independent random variables:{}{}∑∑∑=iiiiiiiYcaYcYa2,σσReview of Inferential Statistics January 21, 20045Alternative Linear Estimators of the Mean321332123211414141ˆ412141ˆ313131ˆXXXXXXXXX++=++=++=µµµExpected Values of Estimators{}{}{}++=++=++=321332123211414141ˆ412141ˆ313131ˆXXXEEXXXEEXXXEEµµµ{}{}{}++=++=++=321332123211414141ˆ412141ˆ313131ˆXEXEXEEXEXEXEEXEXEXEEµµµExpected Values-continued{} {} {} {}{}{} {} {}{}{} {} {}321332123211414141ˆ412141ˆ313131ˆXEXEXEEXEXEXEEXEXEXEE++=++=++=µµµ{}{}{}µµµµµµµµµµµµµµµ43414141ˆ412141ˆ313131ˆ321=++==++==++=EEEReview of Inferential Statistics January 21, 20046Variances of Estimators{} {} {} {}{}{} {} {}{}{} {} {}322212321232322212321222322212321212161161161414141ˆ16141161412141ˆ919191313131ˆXXXXXXXXXXXXXXXXXXσσσσµσσσσσµσσσσσµσ++=++=++=++=++=++={} {} {} {} {}{}{} {} {} {}{}{} {} {} {}XXXXXXXXXXXX222232222222222212163161161161ˆ8316141161ˆ31919191ˆσσσσµσσσσσµσσσσσµσ=++==++==++=Mean Square Error{} ( ){}2ˆˆθθθ−= EMSE{}{}{}θθσθˆˆˆ22BiasMSE +=10=µ{}102=Xσ{}333.301031ˆ1=+=µMSE{}75.301083ˆ2=+=µMSE{}125.8250.6875.110431010163ˆ23=+=−+=µMSEInterval Estimation{} {}()95.96.196.1 =+<<− XXXPσµσµ()95.=≤≤highlowCCPθ{} {}()95.96.196.1 =−>>+ XXXXPσµσPivot method:For the mean:{})1,0(~ norXXσµ−Review of Inferential Statistics January 21, 20047Interval Estimate for Variance{}()1ˆ22−−=∑nXXXσ{}( ){}2)1(22~1ˆ−−×nXnXχσσ{}( ){}95.1ˆ2975),.1(222025),.1(=≥−×≥−− nnXnXPχσσχ{}( ) {}( )95.1ˆ1ˆ2025),.1(222975),.1(2=−×≥≥−×−− nXnnXnXPχσσχσExample of Interval for Variance36.392412141.12241212×>>×FTσn = 25s{FT} = 11s2{FT} = 121df = 2439.36224,.025=χ12.41224,.975=χ{}( ) {}( )95.1ˆ1ˆ2025),.1(222975),.1(2=−×≥≥−×−− nXnnXnXPχσσχσ59.830.15or78.7304.2342>>>>FTFTσσMaximum Likelihood Estimation);(θYf()∏==niinYfYYYf121;),...,,(θ()∏==niisYfYf1;);(θθReview of Inferential Statistics January 21, 20048ML-continued I()()()∏∏====niiniisfYfYfYf11,;);(θθθθ()()()θθθ;,ssYffYf ×=or()()()sssYfYfYf ;,θθ×=ML-continued II()()()sssYfYfYfθθ,; =()() ( )()sssYfYffYfθθθ;;×=()()()()θθθ;;sssYfYffYf ×=ML-continued III()()θθ;~;ssYfYf()() ()∏===niisYfLYf1;;θθθ()[]()[]∑==niiYfL1;loglogθθReview of Inferential Statistics January 21, 20049ML for binomial() ( ) ( )snssnssnL−−−−=πππππ1~1()[] [] [ ] []ππππ−−−+= 1log]1logloglog snLLog-Likelihood:Stata Program to Estimate Binomial Likelihoodversion 8.2************************************************************* Do file to estimate a maximum likeihood for a binomial * Programmed by Bert Kritzer, January 8, 2004************************************************************drop _all* Generate values for piset obs 39egen pi=fill(0.025[.025].975)* compute likelihood for each value of pi (N=20, s=12)* note that binomial coefficient (20!)/(12! x 8!) is constant, * and can be omittedgen LikeLi=(pi^12)*((1-pi)^8)* plot the resulting likelihoodtwoway (connected LikeLi pi, msymbol(none)), ytitle(Likelihood) title(Estimating a Binomial Maximum Likelihood)0 5.00 e-07 1.00 e-06 1.50 e-06Likelihood0 .2 .4 .6 .8 1piEstimating a Binomial Maximum