To find the empirical credibility estimator, the structural parametersofZjin (9.8) need to be estimated.In general there is no explicit for-mula forμTj(Θj) and forV(Tj|Θj).SoμTj(Θj) is replaced by the asymp-totic expectation ofTjandV(Tj|Θj) byn-1times the asymptotic varianceV(Tj,FX|Θj) as in Chapter 9. Therefore we obtain the following asymptoticnon–homogeneous linear credibility estimatorˆμj0∼=μT+Zj[Tj-μT],whereZj=aTns2T+aTnandaT=VbracketleftbigTj(FX|Θj)bracketrightbig,s2T=EbracketleftbigV(Tj,FX|Θj)bracketrightbig,μT=EbracketleftbigμTj(Θj)bracketrightbig=E(Tj).To complete the derivation ofμX(Θj), estimators of the unknown structuralparametersμxs,μT,aTands2Tare needed. The M–estimatorTjcan be rewrit-ten asTj=1nnsummationdisplayr=1Tjr,withTjr= min(Xjr,2Tj).
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106CHAPTER 9.ROBUST STATISTICSWe see from this expression that all losses included in an interval [0,2Tj] willnot be truncated and can be considered as ordinary losses. Denote byTjrthe ordinary portion of a claim amountXSjr=Xjr-Tjrthe excess claim amount.Note that the random variablesTjr,j= 1,...,kandr= 1,...,n, are notconditionally independent given Θj. Therefore an estimator of the asymp-totic varianceV(Tj,FX|Θj) is needed.ReplaceFX|Θjby the empirical dis-tribution of theXjr,r= 1,...,n,j= 1,...,k. After some straightforwardcalculations and a change of normalizing constant fromn-1to (n-1)-1wegetˆs2j=1n-1∑nr=1(Tjr-Tj)2(1-2n∑nr=1IbracketleftbigXjr>2Tjbracketrightbig)2,(9.11)whereIbracketleftbigXjr>2Tjbracketrightbigis an indicator function. Note that the denominator in(9.11) is equal to 1 in the case where allXjr≤2Tj, i.e. in the case whereTj=¯Xj.To summarize, the estimators obtained with the model of B¨uhlmann–Straub with identical weights are given by:ˆμT=1kksummationdisplayj=1Tj,ˆμxs=1kksummationdisplayj=1¯XSj.=1k1nksummationdisplayj=1nsummationdisplayr=1XSjr,ˆs2T=1kksummationdisplayj=1ˆs2j,ˆaT=ksummationdisplayj=1bracketleftbig(Tj-ˆμT)2(k-1)-ˆs2Tknbracketrightbig.Hence the empirical robust credibility estimator is given byˆˆμjX= ˆμxs+ ˆμT+ˆZj[Tj-ˆμT],(9.12)whereˆZj=ˆaTnˆs2T+ ˆaTn.
9.3.*ROBUST CREDIBILITY MODELS107Gisler and Reinhard extend these estimators to allow for different weightswjr. We will not include the detailed derivation here but summarize, in thenext section, the results obtained for Hachemeister’s data set.9.3.3A Numerical IllustrationWe now compare the estimates obtained under B¨uhlmann and B¨uhlmann–Straub models for Hachemeister’s data set (in the presence of outliers) tothose obtained, respectively, with K¨unsch and Gisler–Reinhard’s premiumformulas.
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