03EstimationCb(1)

03EstimationCb(1) - EstimatingEstimableFunctionsof...

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Unformatted text preview: EstimatingEstimableFunctionsof WenowshiftattentionfromE ( y ) totheparametervector . Rememberourt-testquestionsabout 1 2 and y 1 y 2 ? Thosearequestionsabout orlinearcombinationsof . Weveseensomemodelswherethereisauniquesolutionfor sotheelementsof canbeinterpreted.Weveseenothermodels wherethereareaninfinitenumberofsolutionsfor . Thereareothermodelswheresomecomponentsof have uniquesolutionsandothercomponentshaveaninfinitenumberof solutions. Howcanwetellwhenacomponentof isunique? Howcanwetellwhenalinearcombinationof isunique? Whatcanwesayaboutthestatisticalpropertiesofsolutionsfor ? Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 1/19 TheResponseDependson Onlythrough X IntheGauss-MarkovorNormalTheoryGauss-MarkovLinear Model,thedistributionof y dependson onlythrough X ,i.e., y ( X , 2 I ) or y N ( X , 2 I ) If X isnotoffullcolumnrank,thereareinfinitelymanyvectorsin theset { b : Xb = X } foranyfixedvalueof . Thus,nomatterwhatthevalueof E ( y ) ,therewillbeinfinitelymany vectors b suchthat Xb = E ( y ) when X isnotoffullcolumnrank. Theresponsevector y canhelpuslearnabout E ( y )= X ,but when X isnotoffullcolumnrank,thereisnohopeoflearning about aloneunlessadditionalinformationabout isavailable. Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 2/19 TreatmentEffectsModel Rememberthedairycowexperiment:sixexperimentalunitsrandomly assignedtotwotreatments.Onepossiblemodelfortheresponseis calledan effectsmodel . y ij = + i + ij , i = 1 , 2 ; j = 1 , 2 , 3 y 11 y 12 y 13 y 21 y 22 y 23 = + 1 + 1 + 1 + 2 + 2 + 2 + 11 12 13 21 22 23 y 11 y 12 y 13 y 21 y 22 y 23 = 110 110 110 101 101 101 1 2 + 11 12 13 21 22 23 Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 3/19 TreatmentEffectsModel(continued) Inthiscase,itmakesnosensetoestimate =[ , 1 , 2 ] because therearemultiple(infinitelymany,infact)choicesof thatdefine thesamemeanfor y . Forexample, 1 2 = 5 1 1 , 4 6 , or 999 995 993 allyieldsame X = E ( y ) . Whenmultiplevaluesfor definethesame E ( y ) ,wesaythat is non-estimable . Copyright c 2011Dept.ofStatistics(IowaStateUniversity)2011Dept....
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03EstimationCb(1) - EstimatingEstimableFunctionsof...

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