Consider the following estimators for these parameters.Lemma 6.1.LetA,BandCbe matrices andX,Ybe vectors.Underregularity conditions (including proper dimensions):∂∂X(AprimeX)=Aand∂∂X(XprimeB) =B,(6.1)∂∂X(XprimeAX)=2AXand(6.2)∂∂Xbraceleftbig(Y-CX)primeA(Y-CX)bracerightbig=-2CprimeA(Y-CX).(6.3)*Proof:The regularity conditions can be found in any multivariate regres-sion textbook. Left as an exercise.Theorem 6.1.The weighted least squares regression estimators which min-imiseQj(β) =bracketleftbigXj-Yβ(Θj)bracketrightbigprimeV-1jbracketleftbigXj-Yβ(Θj)bracketrightbig,are given byˆβj= (YprimeV-1jY)-1YprimeV-1jXj=UjYprimeV-1jXj,(6.4)where we defineUj= (YprimeV-1jY)-1for notational convenience.
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62CHAPTER 6.CREDIBILITY REGRESSION MODELSProof:By (6.3) we have that∂∂βQj(β) =-2YprimeV-1jbracketleftbigXj-Yβ(Θj)bracketrightbig,then the solutionˆβjmust satisfy the following normal equation:-2YprimeV-1jbracketleftbigXj-Yˆβjbracketrightbig= 0.Hence(YprimeV-1jY)ˆβj=YprimeV-1jXjandˆβj= (YprimeV-1jY)-1YprimeV-1jXj.a50The estimator in (6.4) is expressed in terms ofVj, that is the conditionalcovariance ofXjgiven Θj. The following corollary shows that it can also beexpressed in terms of the unconditional covariance Cov(Xj).Corollary 6.1.LetCj= Cov(Xj) =s2Vj+Y A Yprimethenˆβj= (YprimeC-1jY)-1YprimeC-1jXj.*Proof:see Lemma 6.2 below and the proof of Lemma 1.4.1, page 133 ofGoovaerts et al. (1990).a50Lemma 6.2.(i)E(Xj) =YEbracketleftbigβ(Θj)bracketrightbig=Yb,(ii)E[ˆβj|Θj] =β(Θj) andE[ˆβj] =Ebracketleftbigβ(Θj)bracketrightbig=b,(iii) Cov(Xj) =s2Vj+Y A Yprime=Cj, by definition,(iv) Covbracketleftbigβ(Θj), Xjbracketrightbig=A Yprime,(v) Covbracketleftbigˆβj, β(Θj)bracketrightbig=A,(vi) Cov[ˆβj, Xj] = [A+s2Uj]Yprime, whereUj= (YprimeV-1jY)-1,
6.2.HACHEMEISTER’S REGRESSION MODEL63(vii) Cov[ˆβj] =A+s2Uj, whereUj= (YprimeV-1jY)-1,*Proof:(i) The definition of the modelE(Xj|Θj) =Yβ(Θj) implies thatE(Xj) =EbracketleftbigE(Xj|Θj)bracketrightbig=EbracketleftbigYβ(Θj)bracketrightbig=YEbracketleftbigβ(Θj)bracketrightbig=Yb.(ii)E[ˆβj|Θj]=Ebracketleftbig(YprimeV-1jY)-1YprimeV-1jXj|Θjbracketrightbig=(YprimeV-1jY)-1YprimeV-1jE(Xj|Θj)=(YprimeV-1jY)-1YprimeV-1jYβ(Θj) =β(Θj),with (i) it implies thatE[ˆβj] =EbracketleftbigE(ˆβj|Θj)bracketrightbig=Ebracketleftbigβ(Θj)bracketrightbig.(iii)Cj= Cov(Xj)=EbracketleftbigCov(Xj|Θj)bracketrightbig+ CovbracketleftbigE(Xj|Θj)bracketrightbig=Ebracketleftbigσ2(Θj)Vjbracketrightbig+YCovbracketleftbigβ(Θj)bracketrightbigYprime=s2Vj+Y A Yprime.
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