All classes especially Class 1 exhibit strong trends over time It is thus

All classes especially class 1 exhibit strong trends

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All classes, especially Class 1, exhibit strong trends over time. It is thus natural that the above premiums (the intercept values) be larger that those obtained with B¨uhlmann–Straub’s model. The choice of credibility model is thus very important. A good fit to data should always be seeked once the estimated parameter values have been obtained. Disregarding the trend here could result in a serious under-rating for all classes. Here the collective parameter estimates are: (1) a collective intercept and slope of 1 , 885 and - 32, respectively, (2) an expected variance of ˆ s 2 = 5 . 0 × 10 7 , (3) and variance of conditional means of the regression parameters ˆ A = bracketleftbigg 145 , 359 - 6 , 623 - 6 , 623 302 bracketrightbigg These compare favourably to the B¨uhlmann–Straub’s estimators ˆ m = ¯ X ZW = 1 , 684, ˆ s 2 = 1 . 39 × 10 8 and ˆ a = 89 , 639. Recall that in both models E bracketleftbig Cov( X j | Θ j ) bracketrightbig = E bracketleftbig σ 2 j ) bracketrightbig V j = s 2 w - 1 j 1 0 . . . 0 0 w - 1 j 2 . . . 0 . . . . . . . . . . . . 0 0 . . . w - 1 jn , where the estimator ˆ s 2 of s 2 is more than half smaller in the regression model, so there is less unexplained residual within-class variance left when fitting Hachemeister’s model; that is good.
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68 CHAPTER 6. CREDIBILITY REGRESSION MODELS The comparison of the other variance component, the scalar a = V [ μ (Θ)] in B¨uhlmann–Straub’s model, to the corresponding value in Hachemeister’s model is more complicated; the latter is a n × n diagonal matrix, because the premium changes at each period: Cov bracketleftbig μ j ) bracketrightbig = Cov bracketleftbig Y β j ) bracketrightbig = Y A Y prime . Using the plug–in ˆ A estimators above for Hachemeister’s data set we get the following estimated get ˆ Cov[ μ j )] = Y ˆ A Y prime = 29895 32894 35893 38892 41891 44890 47889 50888 53887 56886 59885 62884 32894 36195 39496 42797 46098 49399 52700 55981 59302 62603 65904 69205 35893 39496 43099 46702 50350 53908 57571 61094 64717 68320 71923 75526 38892 42797 46702 50607 54512 58417 62322 66207 70132 74037 77942 81847 41891 46098 50350 54512 58719 62926 67133 71320 75547 79754 83961 88168 44890 49399 53908 58417 62926 67435 71944 76433 80962 85471 89980 94489 47889 52700 57571 62322 67133 71944 76755 81546 86377 91188 95999 100810 50888 55981 61094 66207 71320 76433 81546 86659 91792 96905 102018 107131 53887 59302 64717 70132 75547 80962 86377 91792 97207 102622 108037 113452 56886 62603 68320 74037 79754 85471 91188 96905 102622 108339 114056 119773 59885 65904 71923 77942 83961 89980 95999 102018 108037 114056 120075 126094 62884 69205 75526 81847 88168 94489 100810 107131 113452 119773 126094 132415 . Compare the diagonal variance values to ˆ a = 89 , 639 in B¨uhlmann–Straub’s model. We see that because of the reverted time scale, the matrix values for the older periods are smaller than ˆ a , but larger than it for the last 4 periods. Remark 6.1. The method could easily be extended to the case when the design matrices Y j change from one class to another. For instance, if some class does not exhibit any inflation, then p could be set to 1 for those classes. One can also model linear inflation (that is p = 2) for some classes and quadratic for others ( p = 3).
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