Now from the above definitions and assumptions we have Cov bracketleftbig μ Θ j

# Now from the above definitions and assumptions we

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. Now from the above definitions and assumptions we have Cov bracketleftbig μ j ) , X i prime r prime bracketrightbig = E braceleftbig Cov bracketleftbig μ j ) , X i prime r prime | Θ j bracketrightbigbracerightbig +Cov braceleftbig E [ μ j ) | Θ j ] , E ( X i prime r prime | Θ j ) bracerightbig , = 0 + Cov braceleftbig μ j ) , E ( X i prime r prime | Θ j ) bracerightbig , (4.4) where E ( X i prime r prime | Θ j ) = braceleftbigg E ( X i prime r prime ) = m , if i prime negationslash = j μ j ) , if i prime = j . Hence, substituting back in (4.4) we get Cov bracketleftbig μ j ) , X i prime r prime bracketrightbig = braceleftbigg Cov braceleftbig μ j ) , m bracerightbig = 0 , if i prime negationslash = j Cov braceleftbig μ j ) , μ j ) bracerightbig = V bracketleftbig μ j ) bracketrightbig = a , if i prime = j = δ i prime j a , where δ ij = braceleftbigg 1 if i = j 0 if i negationslash = j ,

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38 CHAPTER 4. B ¨ UHLMANN’S CREDIBILITY MODELS is Kronecker’s symbol. Similarly Cov( X ir , X i prime r prime ) = E braceleftbig Cov( X ir , X i prime r prime | Θ i ) bracerightbig + Cov braceleftbig E ( X ir | Θ i ) , E ( X i prime r prime | Θ i ) bracerightbig , = δ ii prime ( δ rr prime s 2 + a ) , which allows us to re–write (4.3) as δ i prime j a = k summationdisplay i =1 n summationdisplay r =1 c j ir δ ii prime ( a + δ rr prime s 2 ) , = n summationdisplay r =1 c j i prime r ( a + δ rr prime s 2 ) , (4.5) = a c j i prime . + c j i prime r prime s 2 , (4.6) for every i prime = 1 , . . . , k and r prime = 1 , . . . , n, where c j i prime . = n r =1 c j i prime r . Summing on both sides with respect to r prime we obtain δ i prime j a n = a c j i prime . n + s 2 c j i prime . , for every i prime = 1 , . . . , k, that is c j i prime . = δ i prime j a n s 2 + a n = δ i prime j Z , where Z = a n s 2 + a n . A substitution of this final expression in (4.6) gives c j i prime r prime = δ i prime j (1 - Z ) a s 2 , for every i prime = 1 , . . . , k and r prime = 1 , . . . , n . We see that all these coefficients are 0 unless i prime = j and that the optimal estimator is thus c j 0 + k summationdisplay i =1 n summationdisplay r =1 c j ir X ir = s 2 s 2 + a n m + n summationdisplay r =1 [(1 - Z ) a s 2 X jr ] = (1 - Z ) m + Z ¯ X j a50 Although called a credibility “estimator”, the above premium formula depends on the portfolio parameters m , s 2 and a , which have yet to be estimated by some distribution free approach.
4.1. THE CLASSICAL MODEL OF B ¨ UHLMANN (1967) 39 Lemma 4.1. The following estimators ˆ m = ¯ X = 1 k k summationdisplay j =1 ¯ X j = 1 kn k summationdisplay j =1 n summationdisplay r =1 X jr , ˆ s 2 = 1 k ( n - 1) k summationdisplay j =1 n summationdisplay r =1 ( X jr - ¯ X j ) 2 , and ˆ a = 1 ( k - 1) k summationdisplay j =1 ( ¯ X j - ¯ X ) 2 - 1 n ˆ s 2 , are unbiased for m , s 2 and a , respectively. Proof: [De Vylder (1981)] The first assertion E ( ˆ m ) = m is trivial. Then, from classical statistics for iid observations it is clear that for every fixed j = 1 , . . . , k . 1 ( n - 1) n summationdisplay r =1 ( X jr - ¯ X j ) 2 is a conditionally unbiased estimator of σ 2 j ) = V ( X jr | Θ j ), which means that 0 = E bracketleftBig 1 ( n - 1) n summationdisplay r =1 ( X jr - ¯ X j ) 2 - σ 2 j ) | Θ j bracketrightBig .

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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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