X j X 1 k n n 1 k j 1 n r 1 ˆ IF X jr T j X jr X j 1 k 1 k j 1 T j T 2 where ˆ

X j x 1 k n n 1 k j 1 n r 1 ˆ if x jr t j x jr x j 1

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) ( ¯ X j - ¯ X . ) - 1 k n ( n - 1) k j =1 n r =1 ˆ IF ( X jr ; T j ) ( X jr - ¯ X j ) 1 k - 1 k j =1 ( T j - ¯ T . ) 2 , where ˆ IF ( x ; T j ) = parenleftBig x T j parenrightBig T 2 j n r =1 X jr I bracketleftbig (1 - c 1 ) T j X jr (1 + c 2 ) T j bracketrightbig . Notice again the use of non-robust sample averages as commented in Remark (V) above. 9.3.2 Gisler and Reinhard’s Model Definitions K¨unsch uses robust location estimators T j = T j ( X j 1 ,...,X jn ) to replace the usual contract averages ¯ X j. , since these perform reasonably well in the neigh- borhood of the true model. Gisler and Reinhard (1993) propose to divide the pure risk premium into two components: an ordinary–part for average claims, and an excess–part for outlying claims, which can be estimated separately. More precisely, μ X j ) = μ 0 j ) + μ xs j ) ,
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104 CHAPTER 9. ROBUST STATISTICS where μ X j ) = E ( X jr | Θ j ), μ 0 j ) denotes the ordinary part and μ xs j ) is the excess part. The ordinary part μ 0 j ) is generated by the claims due to ordinary losses, whereas the excess–part μ xs j ) is the additional expected claims generated mainly by extraordinary events such as large fires or hurricanes. The excess–part is the one that usually generates outlying observations. To estimate the ordinary part μ 0 j ), credibility and robust statistics are combined, i.e. a credibility estimator based on a robust statistic T j = T j ( X j 1 ,...,X jn ), for j = 1 ,...,k is used. By definition μ 0 j ) = E ( T j | Θ j ) . All risks in the portfolio are assumed equally exposed to outlier events. This can be expressed in the following manner: μ xs j ) = μ xs , j = 1 ,...,k. Note that if an a–priori assumption can be made to establish how certain risks are more exposed to outlier events than others, then one can always define a known matrix A k × 1 such that μ xs j ) = A μ xs . The robust credibility estimator of μ X j ) is given by ˆ μ j X = ˆ μ j 0 + μ xs , (9.5) where ˆ μ j 0 is estimated by standard robust techniques, without regard to bias: ˆ μ 0 = E ( T j ) + Z j [ T j - E ( T j )] (9.6) = μ T j + Z j [ T j - μ T j ] (9.7) and Z j = V bracketleftbig E ( T j | Θ j ) bracketrightbig E bracketleftbig V ( T j | Θ j ) bracketrightbig + V bracketleftbig E ( T j | Θ j ) bracketrightbig = V bracketleftbig μ T j j )] E bracketleftbig V ( T j | Θ j ) bracketrightbig + V bracketleftbig μ T j j ) bracketrightbig . (9.8) Weighted model with identical volumes To simplify the derivation, first consider B¨uhlmann–Straub’s model with identical weights, that is (i) The contracts j = 1 ,...,k [i.e. the pair vectors (Θ j ,X j )] are independent and the Θ j variables are identically distributed;
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9.3. *ROBUST CREDIBILITY MODELS 105 (ii) r,s = 1 ,...,n and j = 1 ,...,k , E ( X jr | Θ j ) = μ j ) , Cov( X jr ,X js | Θ j ) = δ rs w σ 2 j ) , where w is a known weight. The M–estimator is defined implicitly by n summationdisplay r =1 min( X jr T j - 1 , 1) = 0 , (9.9) which can be rewritten as T j = 1 n n summationdisplay r =1 min( X jr , 2 T j ) . (9.10) An algorithmic solution to (9.10) is suggested by the authors to calculate T j .
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