The credibility estimator of μ Θ j E X jr Θ j proposed by Kunsch in Case I is

The credibility estimator of μ θ j e x jr θ j

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The credibility estimator of μ j ) = E [ X jr | Θ j ] proposed by K¨unsch in Case I is inspired by B¨uhlmann’s non–homogeneous linear Bayes rule M I j = m + Z bracketleftbig T j - E ( T j ) bracketrightbig , (9.1) and where m = E bracketleftbig μ j ) bracketrightbig = integraldisplay Ω μ ( θ ) dU ( θ ) , is the known portfolio mean and T j = T j ( X j 1 ,...,X jn ) is an M–estimator defined implicitly as the solution of n summationdisplay r =1 χ parenleftbigg X jr T j parenrightbigg = 0 , (9.2) where χ ( z ) = max {- c 1 , min( z - 1 ,c 2 ) } , for 0 <c 1 1 and 0 <c 2 . The following alternative formulation to (9.2) 1 n n summationdisplay r =1 ˜ χ parenleftbigg X jr T j parenrightbigg = 1 ,
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102 CHAPTER 9. ROBUST STATISTICS where ˜ χ ( z ) = max { 1 - c 1 , min( z, 1 + c 2 ) } , allows a recursive solution of this normal equation: T ( m +1) j = bracketleftBigg 1 n n summationdisplay r =1 ˜ χ parenleftBigg X jr T ( m ) j parenrightBiggbracketrightBigg 1 2 T ( m ) j , for m 0 , with a robust starting value of T 0 j = median { X j 1 ,...,X jn } . Its convergence follows from Huber (1981), Section 8.6. The credibility factor Z suggested by K¨unsch in Case I is also a robust version of B¨uhlmann’s linear Bayes rule: Z = Cov bracketleftbig E ( T j | Θ j ) j ) bracketrightbig E bracketleftbig V ( T j | Θ j ) bracketrightbig + V bracketleftbig E ( T j | Θ j ) bracketrightbig , (9.3) whose exact value can, in theory, be calculated from U and F X | Θ . Remarks: (I) M I j is scale invariant, i.e. if all X jr are multiplied by a constant c then so is M I j . (II) It reproduces B¨uhlmann’s estimator if c 1 = 1 and c 2 = . (III) It is unbiased. (IV) Under this pure Bayesian approach, the exact credibility premium E bracketleftbig μ j ) | X j 1 ,...,X jn bracketrightbig could be computed; it might be non–linear but possibly robust. To ensure linearity, K¨unsch suggests instead to use a non–robust “optimal” linear estimator and to robustify it. However, this destroys the optimality. In Case II, the portfolio mean m is unknown and so is E ( T j ). Using a semi–parametric approach K¨unsch suggests to estimate these by their corre- sponding sample averages. This produces the following credibility premium: M II j = ¯ X . + Z [ T j - ¯ T . ] , (9.4) where ¯ X . = 1 kn k summationdisplay j =1 n summationdisplay r =1 X jr and ¯ T . = 1 k k summationdisplay j =1 T j . Remarks: (I) and (II) above also apply in Case II. In addition
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9.3. *ROBUST CREDIBILITY MODELS 103 (V) To achieve unbiasedness, the non–robust estimator ¯ X . is used for the portfolio mean. From a robust point of view, the credibility estimator (1 - Z ) ¯ T . + ZT j is preferable, but the later is biased. (VI) It can be seen that the influence function of the functional T used here in (9.2) is given by IF ( x ; T ) = χ parenleftBig x T parenrightBig T 2 M - 1 , where M = integraldisplay χ prime parenleftBig x T parenrightBig xdF X | Θ ( x ) . The empirical credibility estimator for (9.4) is then obtained by replacing the parameter in (9.3) with robust sample versions obtained through the estimated influence function ˆ Z = 1 k - 1 k j =1 ( T j - ¯ T .
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