03EstimationCb(1)

03EstimationCb(1) - β y β Rememberourt-testquestionsabout...

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

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