To find the empirical credibility estimator the structural parameters of Z j in

To find the empirical credibility estimator the

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To find the empirical credibility estimator, the structural parameters of Z j in (9.8) need to be estimated. In general there is no explicit for- mula for μ T j j ) and for V ( T j | Θ j ). So μ T j j ) is replaced by the asymp- totic expectation of T j and V ( T j | Θ j ) by n - 1 times the asymptotic variance V ( T j ,F X | Θ j ) as in Chapter 9. Therefore we obtain the following asymptotic non–homogeneous linear credibility estimator ˆ μ j 0 = μ T + Z j [ T j - μ T ] , where Z j = a T n s 2 T + a T n and a T = V bracketleftbig T j ( F X | Θ j ) bracketrightbig , s 2 T = E bracketleftbig V ( T j ,F X | Θ j ) bracketrightbig , μ T = E bracketleftbig μ T j j ) bracketrightbig = E ( T j ) . To complete the derivation of μ X j ), estimators of the unknown structural parameters μ xs T ,a T and s 2 T are needed. The M–estimator T j can be rewrit- ten as T j = 1 n n summationdisplay r =1 T jr , with T jr = min( X jr , 2 T j ) .
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106 CHAPTER 9. ROBUST STATISTICS We see from this expression that all losses included in an interval [0 , 2 T j ] will not be truncated and can be considered as ordinary losses. Denote by T jr the ordinary portion of a claim amount XS jr = X jr - T jr the excess claim amount . Note that the random variables T jr , j = 1 ,...,k and r = 1 ,...,n , are not conditionally independent given Θ j . Therefore an estimator of the asymp- totic variance V ( T j ,F X | Θ j ) is needed. Replace F X | Θ j by the empirical dis- tribution of the X jr , r = 1 ,...,n , j = 1 ,...,k . After some straightforward calculations and a change of normalizing constant from n - 1 to ( n - 1) - 1 we get ˆ s 2 j = 1 n - 1 n r =1 ( T jr - T j ) 2 (1 - 2 n n r =1 I bracketleftbig X jr > 2 T j bracketrightbig ) 2 , (9.11) where I bracketleftbig X jr > 2 T j bracketrightbig is an indicator function. Note that the denominator in (9.11) is equal to 1 in the case where all X jr 2 T j , i.e. in the case where T j = ¯ X j . To summarize, the estimators obtained with the model of B¨uhlmann– Straub with identical weights are given by: ˆ μ T = 1 k k summationdisplay j =1 T j , ˆ μ xs = 1 k k summationdisplay j =1 ¯ XS j. = 1 k 1 n k summationdisplay j =1 n summationdisplay r =1 XS jr , ˆ s 2 T = 1 k k summationdisplay j =1 ˆ s 2 j , ˆ a T = k summationdisplay j =1 bracketleftbig ( T j - ˆ μ T ) 2 ( k - 1) - ˆ s 2 T kn bracketrightbig . Hence the empirical robust credibility estimator is given by ˆ ˆ μ j X = ˆ μ xs + ˆ μ T + ˆ Z j [ T j - ˆ μ T ] , (9.12) where ˆ Z j = ˆ a T n ˆ s 2 T + ˆ a T n .
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9.3. *ROBUST CREDIBILITY MODELS 107 Gisler and Reinhard extend these estimators to allow for different weights w jr . We will not include the detailed derivation here but summarize, in the next section, the results obtained for Hachemeister’s data set. 9.3.3 A Numerical Illustration We now compare the estimates obtained under B¨uhlmann and B¨uhlmann– Straub models for Hachemeister’s data set (in the presence of outliers) to those obtained, respectively, with K¨unsch and Gisler–Reinhard’s premium formulas.
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