triangle 42 The BuhlmannStraub Model First published in German Buhlmann and

# Triangle 42 the buhlmannstraub model first published

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triangle 4.2 The B¨uhlmann–Straub Model First published in German, B¨uhlmann and Straub (1970) generalizes the classical model by the introduction of weights. The model assumptions are: (i) (Θ j , X j ) prime are (pair–wise) independent, and the Θ j ’s are identically dis- tributed for j = 1 , . . . , k ,

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42 CHAPTER 4. B ¨ UHLMANN’S CREDIBILITY MODELS (ii) for each fixed j , the E ( X jr | Θ j ) = μ j ) < and Cov( X jr , X ju | Θ j ) = δ ru w jr σ 2 j ), for all r = 1 , . . . , n , where the w jr are known weights associated with the observations X jr and δ ru is Kronecker’s symbol. Note here how the conditional variances V ( X jr | Θ j ) = σ 2 j ) /w jr vary here from one observation to the other due to the individual weights. The following notations and covariance relations will be used to derive the credibility estimator for B¨uhlmann–Straub’s model (B–S): w .. = k summationdisplay j =1 w j. = k summationdisplay j =1 n summationdisplay r =1 w jr , Z . = k summationdisplay j =1 Z j , where Z j = a w j. s 2 + a w j. , ¯ X jW = n summationdisplay r =1 w jr w j. X jr and ¯ X WW = k summationdisplay j =1 w j. w .. ¯ X jW , ¯ X ZW = k summationdisplay j =1 Z j Z . ¯ X jW . Lemma 4.2. For a fixed class j = 1 , . . . , k : (1) Cov[ μ j ) , X ir ] = δ ij a , for any i = 1 , . . . , k and r = 1 , . . . , n , (2) Cov[ X jr , X iu ] = δ ij ( a + δ ru s 2 w jr ) , for any i, j, r, u , (3) Cov[ X jr , ¯ X iW ] = δ ij Cov[ ¯ X jW , ¯ X jW ] = δ ij V ( ¯ X jW ) = δ ij ( a + s 2 w j. ) , for any i, j, r , (4) Cov[ ¯ X jW , ¯ X ZW ] = Cov[ ¯ X ZW , ¯ X ZW ] = V ( ¯ X ZW ) = a Z . , (5) Cov[ ¯ X jW , ¯ X WW ] = s 2 w .. + a w j. w .. , (6) Cov[ ¯ X WW , ¯ X WW ] = V ( ¯ X WW ) = s 2 w .. + a j ( w j. w .. ) 2 . Remark 4.3. Note that E ( X ir | Θ j ) = braceleftbigg μ j ) if i = j E ( X ir ) = m if i negationslash = j , (4.7)
4.2. THE B ¨ UHLMANN–STRAUB MODEL 43 while E ( X jr ) = E bracketleftbig E ( X jr | Θ j ) bracketrightbig = E bracketleftbig μ j ) bracketrightbig = m implies that E ( ¯ X jW ) = E ( ¯ X WW ) = E ( ¯ X ZW ) = m . They are all portfolio–unbiased estimators of m . Remark 4.4. The above lemma gives the variance of all the weighted aver- ages needed for the portfolio: Lemma 4.2-(2) V ( X jr ) = a + s 2 w jr Lemma 4.2-(3) V ( ¯ X jW ) = a + s 2 w j. Lemma 4.2-(4) V ( ¯ X ZW ) = a Z . Lemma 4.2-(6) V ( ¯ X WW ) = a summationdisplay j ( w j. w .. ) 2 + s 2 w .. . Proof of Lemma 4.2: (1) Cov[ μ j ) , X ir ] = E braceleftbig Cov bracketleftbig μ j ) , X ir | Θ j bracketrightbigbracerightbig +Cov braceleftbig E bracketleftbig μ j ) | Θ j bracketrightbig , E ( X ir | Θ j ) bracerightbig , = 0 + Cov bracketleftbig μ j ) , E ( X ir | Θ j ) bracketrightbig , where E ( X ir | Θ j ) = δ ij μ j ) + (1 - δ ij ) m , by (4.7). Hence Cov bracketleftbig μ j ) , X ir bracketrightbig = δ ij V [ μ j )] + 0 = δ ij a . (2) Similarly for Cov( X jr , X iu ). Using the iterated law for conditional co- variances it follows that Cov( X jr , X iu ) = E bracketleftbig Cov( X jr , X iu | Θ j ) bracketrightbig +Cov bracketleftbig E ( X jr | Θ j ) , E ( X iu | Θ j ) bracketrightbig , where in the first term, Cov( X jr , X iu | Θ j ) = δ ij Cov( X jr , X ju | Θ j ) = δ ij δ ru σ 2 j ) /w jr , by assumption (ii). Again in the second term, E ( X iu | Θ j ) = δ ij μ j ) + (1 - δ ij ) m . Hence, substituting we get Cov( X jr , X iu ) = δ ij δ ru w jr E bracketleftbig σ 2 j ) bracketrightbig bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright

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• Fall '09
• Dr.D.Dryanov
• Trigraph, Estimation theory, Mean squared error, Bias of an estimator, Credibility Models

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