triangle 42 The BuhlmannStraub Model First published in German Buhlmann and

Triangle 42 the buhlmannstraub model first published

This preview shows page 7 - 10 out of 15 pages.

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 ,
Image of page 7

Subscribe to view the full document.

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)
Image of page 8
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
Image of page 9

Subscribe to view the full document.

Image of page 10
  • Fall '09
  • Dr.D.Dryanov
  • Trigraph, Estimation theory, Mean squared error, Bias of an estimator, Credibility Models

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes