(ii) For the mean we havensummationdisplayi=1ψ(Xi,t) =nsummationdisplayi=1(Xi-t) = 0.
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9.2.ROBUST INFERENCE METHODS99(iii) For the maximum likelihood estimator of a family of densitiesfθtheestimating equation is given bynsummationdisplayi=1ψ(Xi,θ) =nsummationdisplayi=1∂∂θ[lnfθ(Xi)] =nsummationdisplayi=1s(Xi,θ) = 0.*Theorem 9.1.LetTnbe an M–estimator andIF(x;T,Fθ) the influencefunction ofTatFθthenIF(x;T,Fθ)=ψ[x,T(Fθ)]-integraltext∂∂θ[ψ(x,θ)]T(Fθ)dFθ(x)=-ψ[x,T(Fθ)]∂∂θ[integraltextψ(x,θ)dFθ(x)]T(Fθ).The above result implies thatIF(x;T,Fθ) is directly proportional toψ[x,T(Fθ)]. For instance:(i) ifψ(x,t) =x-tthe influence function (of the mean) is given byIF(x;T,Fθ) =x-T(Fθ), which is not bounded inx,(ii) ifψ(x,t) = sign(x-t) the influence function (of the median) is givenbyIF(x;T,Fθ) =sign[x-T(Fθ)]2fθ[T(Fθ)],which is bounded inx.This means that the desired shape of the influence function is achievedby an appropriate selection of the functionψ.*Corollary 1.Consider the distribution functionFθand its correspondingdensityfθ, whereθ∈Ω. Assuming thatT(Fθ) is Fisher consistent, i.e.integraldisplay∞-∞ψ[x,T(Fθ)]dFθ(x) = 0,or, equivalently,T(Fθ) =θ. Then, by integration by parts, we have thatIF(x;T,Fθ) =ψ(x,θ)integraltextψ(x,θ)s(x,θ)dFθ(x),wheres(x,θ) =∂∂θln[fθ(x)]is the score function, and no derivatives ofψare required.
100CHAPTER 9.ROBUST STATISTICSWith the above result we can conclude thatφ(x,θ) =IF(x;T,Fθ) forsome Fisher consistent M–functional if and only ifintegraldisplayφ(x,θ)dFθ(x) = 0andintegraldisplayφ(x,θ)s(x,θ)dFθ(x) = 126.96.36.199*Optimal Robust EstimatorsFrom the above discussion it seems reasonnable to seek Fisher–consistentstatisticsTwhich minimize the asymptotic variance, subject to a limitLonthe gross–error sensitivity.In other words, we need to find aφ(x,θ) suchthat(i)integraltextφ(x,θ)dFθ(x) = 0,(ii)integraltextφ(x,θ)s(x,θ)dFθ(x) = 1,(iii)|φ(x,θ)|2Rφ(x,θ)2dFθ(x)≤L2,which will minimizeintegraldisplayφ(x,θ)2dFθ(x).Denote[y]cb=cify≥cyifb≤y≤cbify≤b.Then the solution is given byφ(x,θ) =[s(x,θ)-a(θ)]b(θ)-b(θ)M(θ),whereM(θ) =integraldisplay[s(x,θ)-a(θ)]b(θ)-b(θ)s(x,θ)dFθ(x)anda(θ),b(θ) are such that the other constraints are satisfied (which can bedone ifL≥1).
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9.3.*ROBUST CREDIBILITY MODELS1019.3*Robust Credibility Models9.3.1K¨unsch’s ModelK¨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) (Θj,Xj)primeare i.i.d.,(ii) Θjis distributed according toU,(iii) given Θj, the claimsXj1,...,Xjnare conditionally i.i.d. with distribu-tionFX|Θ,where he distinguishes:Case I:UandFX|Θare known (Bayesian approach),Case II: the above prior and conditional distributions are unknown anda semi–parametric approach is used.
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