iv Cov bracketleftbig β Θ j X j bracketrightbig E braceleftbig Cov

Iv cov bracketleftbig β θ j x j bracketrightbig e

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(iv) Cov bracketleftbig β j ) , X j bracketrightbig = E braceleftbig Cov bracketleftbig β j ) , X j | Θ j bracketrightbigbracerightbig + Cov braceleftbig E bracketleftbig β j ) | Θ j bracketrightbig , E [ X j | Θ j ] bracerightbig = E (0) + Cov bracketleftbig β j ) , Y β j ) bracketrightbig = A Y prime . (v) Cov bracketleftbig ˆ β j , β j ) bracketrightbig = Cov bracketleftbig ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j , β j ) bracketrightbig = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j Cov bracketleftbig X j , β j ) bracketrightbig = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j ( Cov bracketleftbig β j ) , X j bracketrightbig) prime = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j ( Y prime ) prime A prime = A .
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64 CHAPTER 6. CREDIBILITY REGRESSION MODELS (vi) Cov[ ˆ β j , X j ] = Cov[( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j , X j ] = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j Cov( X j ) = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j ( s 2 V j + Y A Y prime ) = s 2 ( Y prime V - 1 j Y ) - 1 Y prime + A Y prime = s 2 U j Y prime + A Y prime . (vii) Cov bracketleftbig ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j X j bracketrightbig = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j Cov( X j ) V - 1 j Y bracketleftbig ( Y prime V - 1 j Y ) - 1 ] prime = ( Y prime V - 1 j Y ) - 1 Y prime V - 1 j ( s 2 V j + Y A Y prime ) V - 1 j Y ( Y prime V - 1 j Y ) - 1 = s 2 ( Y prime V - 1 j Y ) - 1 + A = s 2 ( Y prime V - 1 j Y ) - 1 + A = s 2 U j + A . a50 Theorem 6.2. The non–homogeneous linear regression credibility estimator B a j of β j ) that minimises E braceleftbigbracketleftbig β j ) - B a j bracketrightbig prime Σ j bracketleftbig β j ) - B a j bracketrightbigbracerightbig is given by B a j = Z j ˆ β j + ( I - Z j ) b , where Z j = A ( A + s 2 U j ) - 1 = Cov bracketleftbig β j ) bracketrightbig bracketleftbig Cov( ˆ β j ) bracketrightbig - 1 . (6.5) Hence the non–homogeneous linear regression credibility estimator M a j of μ j ) is given by M a j = Y B a j . *Proof : Let g j = c 0 + c 1 X 0 j be an arbitrary vector that is a non–homogeneous linear combination of the X j and epsilon1 0. Then for any positive definite matrix Σ j , consider d j ( epsilon1 ) = E braceleftbigbracketleftbig β j ) - B a j - epsilon1g j bracketrightbig prime Σ j bracketleftbig β j ) - B a j - epsilon1 g j bracketrightbigbracerightbig .
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6.2. HACHEMEISTER’S REGRESSION MODEL 65 The theorem holds if d prime j (0) = 0 for any g j . Now, by Lemma 6.1, d prime j ( epsilon1 ) = - 2 E braceleftbig g prime j Σ j bracketleftbig β j ) - B a j bracketrightbigbracerightbig + 2 epsilon1 E bracketleftbig g prime j Σ j g j bracketrightbig = - 2 E braceleftbig g prime j Σ j bracketleftbig β j ) - B a j - epsilon1g j bracketrightbigbracerightbig . (6.6) Then define the following reduced variables β 0 j ) = β j ) - b , ˆ β 0 j = ˆ β j - b , X 0 j = X j - Y b j , so that our normal equation (6.6) at epsilon1 = 0 reduces to E braceleftbig g prime j Σ j bracketleftbig β 0 j ) - Z j ˆ β 0 j bracketrightbigbracerightbig = 0 . Now since g j = c 0 + c 1 X 0 j , equivalently we need to prove that E braceleftbigbracketleftbig c prime 0 + ( X 0 j ) prime c prime 1 bracketrightbig Σ j bracketleftbig β 0 j ) - Z j ˆ β 0 j bracketrightbigbracerightbig = 0 .
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