ii For the mean we have n summationdisplay i 1 ψ X i t n summationdisplay i 1 X

Ii for the mean we have n summationdisplay i 1 ψ x i

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(ii) For the mean we have n summationdisplay i =1 ψ ( X i ,t ) = n summationdisplay i =1 ( X i - t ) = 0 .
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9.2. ROBUST INFERENCE METHODS 99 (iii) For the maximum likelihood estimator of a family of densities f θ the estimating equation is given by n summationdisplay i =1 ψ ( X i ) = n summationdisplay i =1 ∂θ [ln f θ ( X i )] = n summationdisplay i =1 s ( X i ) = 0 . *Theorem 9.1. Let T n be an M–estimator and IF ( x ; T,F θ ) the influence function of T at F θ then IF ( x ; T,F θ ) = ψ [ x,T ( F θ )] - integraltext ∂θ [ ψ ( x,θ )] T ( F θ ) dF θ ( x ) = - ψ [ x,T ( F θ )] ∂θ [ integraltext ψ ( x,θ ) dF θ ( x )] T ( F θ ) . The above result implies that IF ( x ; T,F θ ) is directly proportional to ψ [ x,T ( F θ )]. For instance: (i) if ψ ( x,t ) = x - t the influence function (of the mean) is given by IF ( x ; T,F θ ) = x - T ( F θ ), which is not bounded in x , (ii) if ψ ( x,t ) = sign( x - t ) the influence function (of the median) is given by IF ( x ; T,F θ ) = sign[ x - T ( F θ )] 2 f θ [ T ( F θ )] , which is bounded in x . This means that the desired shape of the influence function is achieved by an appropriate selection of the function ψ . *Corollary 1. Consider the distribution function F θ and its corresponding density f θ , where θ Ω. Assuming that T ( F θ ) is Fisher consistent, i.e. integraldisplay -∞ ψ [ x,T ( F θ )] dF θ ( x ) = 0 , or, equivalently, T ( F θ ) = θ . Then, by integration by parts, we have that IF ( x ; T,F θ ) = ψ ( x,θ ) integraltext ψ ( x,θ ) s ( x,θ ) dF θ ( x ) , where s ( x,θ ) = ∂θ ln[ f θ ( x )] is the score function, and no derivatives of ψ are required.
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100 CHAPTER 9. ROBUST STATISTICS With the above result we can conclude that φ ( x,θ ) = IF ( x ; T,F θ ) for some Fisher consistent M–functional if and only if integraldisplay φ ( x,θ ) dF θ ( x ) = 0 and integraldisplay φ ( x,θ ) s ( x,θ ) dF θ ( x ) = 1 . 9.2.3 *Optimal Robust Estimators From the above discussion it seems reasonnable to seek Fisher–consistent statistics T which minimize the asymptotic variance, subject to a limit L on the gross–error sensitivity. In other words, we need to find a φ ( x,θ ) such that (i) integraltext φ ( x,θ ) dF θ ( x ) = 0, (ii) integraltext φ ( x,θ ) s ( x,θ ) dF θ ( x ) = 1, (iii) | φ ( x,θ ) | 2 R φ ( x,θ ) 2 dF θ ( x ) L 2 , which will minimize integraldisplay φ ( x,θ ) 2 dF θ ( x ) . Denote [ y ] c b = c if y c y if b y c b if y b . Then the solution is given by φ ( x,θ ) = [ s ( x,θ ) - a ( θ )] b ( θ ) - b ( θ ) M ( θ ) , where M ( θ ) = integraldisplay [ s ( x,θ ) - a ( θ )] b ( θ ) - b ( θ ) s ( x,θ ) dF θ ( x ) and a ( θ ), b ( θ ) are such that the other constraints are satisfied (which can be done if L 1).
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9.3. *ROBUST CREDIBILITY MODELS 101 9.3 *Robust Credibility Models 9.3.1 K¨unsch’s Model K¨unsch (1992) robustifies the classical model of B¨uhlmann (1967) by replac- ing the contract sample averages by location M–estimators of Huber’s type. More precisely, he assumes that: (i) (Θ j ,X j ) prime are i.i.d., (ii) Θ j is distributed according to U , (iii) given Θ j , the claims X j 1 ,...,X jn are conditionally i.i.d. with distribu- tion F X | Θ , where he distinguishes: Case I: U and F X | Θ are known (Bayesian approach), Case II: the above prior and conditional distributions are unknown and a semi–parametric approach is used.
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