Table 93 Contaminated Buhlmanns Premiums Outlier 1690 5000 6000 7000 X 1 2064

Table 93 contaminated buhlmanns premiums outlier 1690

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Table 9.3: Contaminated B¨uhlmann’s Premiums Outlier 1,690 5,000 6,000 7,000 ¯ X 1 2,064 2,064 2,064 2,064 ¯ X 2 1,511 1,511 1,511 1,511 ¯ X 3 1,822 1,822 1,822 1,822 ¯ X 4 1,360 1,360 1,360 1,360 ¯ X 5 1,599 1,874 1,958 2,041 Z 0.9496 0.7546 0.6553 0.5521 ˆ μ 1 2,044 1,981 1,953 1,928 ˆ μ 2 1,519 1,563 1,591 1,622 ˆ μ 3 1,814 1,798 1,794 1,794 ˆ μ 4 1,376 1,450 1,493 1,539 ˆ μ 5 1,602 1,838 1,883 1,915 ¯ X 1,671 1,726 1,743 1,760 The results produced by K¨unsch’s model in Case II above, with c 1 = 1 and c 2 = 1 are given by Table 9.4. Notice how the effect of the contamination in Contract 5 is limited in the individual estimator and on the credibility factor Z . Now refer to the contaminated B¨uhlmann–Straub premiums given in the introduction of Chapter 9. We observe that as the outlier value increases, it affects the estimated credibility factor ˆ Z j in a drastic way. As we see in Table 9.5, the introduction of weights in Gisler–Reinhard’s model reduces even further the level of contamination on Z . The results below were produced by Gisler–Reinhard’s model with c 1 = 1 and c 2 = 1.
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108 CHAPTER 9. ROBUST STATISTICS Table 9.4: K¨unsch’s Premiums Outlier 1,690 5,000 6,000 7,000 T 1 2,064 2,064 2,064 2,064 T 2 1,511 1,511 1,511 1,511 T 3 1,822 1,822 1,822 1,822 T 4 1,360 1,360 1,360 1,360 T 5 1,599 1,749 1,749 1,749 Z 0.9496 0.8247 0.7958 0.7668 ˆ μ 1 2,044 2,025 2,031 2,038 ˆ μ 2 1,519 1,569 1,591 1,613 ˆ μ 3 1,814 1,826 1,839 1,852 ˆ μ 4 1,376 1,445 1,472 1,498 ˆ μ 5 1,602 1,766 1,781 1,796 ¯ T . 1,671 1,701 1,701 1,701 When the data from Contract 5 is not contaminated (i.e. when the outlier value is equal to 1 , 690), we obtain the same results as with B¨uhlmann– Straub’s model. Also notice that even if the outlier value increases, the estimated credibility factors ˆ Z j become constant, as the ordinary part is no longer affected by the increase. The estimation for μ T , the ordinary part, becomes constant as well. The only part affected by the increase in the outlier value is the estimation of the excess part, μ XS .
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9.3. *ROBUST CREDIBILITY MODELS 109 Table 9.5: Gisler–Reinhard’s Premiums Outlier 1,690 5,000 6,000 7,000 T 1 2,061 2,061 2,061 2,061 T 2 1,511 1,511 1,511 1,511 T 3 1,806 1,806 1,806 1,806 T 4 1,353 1,353 1,353 1,353 T 5 1,600 1,777 1,777 1,777 ˆ Z 1 0.9848 0.9315 0.9315 0.9315 ˆ Z 2 0.9278 0.7300 0.7300 0.7300 ˆ Z 3 0.8987 0.6511 0.6511 0.6511 ˆ Z 4 0.7283 0.3606 0.3606 0.3606 ˆ Z 5 0.9589 0.8307 0.8307 0.8307 ˆ μ 1 2,055 2,071 2,091 2,111 ˆ μ 2 1,524 1,608 1,628 1,648 ˆ μ 3 1,793 1,820 1,839 1,859 ˆ μ 4 1,443 1,642 1,661 1,681 ˆ μ 5 1,603 1,793 1,813 1,832 ˆ μ XS 0 31.26 50.94 70.62 ˆ μ T 1,684 1,756 1,756 1,756
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