Theorem 71 For a loglink function a normal asymptotic approximation gives π i Φ

Theorem 71 for a loglink function a normal asymptotic

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Theorem 7.1. For a log–link function, a normal asymptotic approximation gives π i = Φ( ln(1 + r ) s i ) - Φ( ln(1 - r ) s i ) , (7.8) where Φ is the standard normal cdf and s i are as in (7.7). *Proof: see Garrido and Zhou (2009). square For the case of a general link function g , let Q 1 = g [(1 - r ) μ i ] - g ( μ i ) , (7.9) Q 2 = g [(1 + r ) μ i ] - g ( μ i ) . (7.10) Theorem 7.2. For a general link function g , an approximate normal asymp- totic result gives π i = Φ( Q 2 s i ) - Φ( Q 1 s i ) , (7.11) where Φ is the standard normal cdf and s i as above. For a GLM model, these ˆ π i can be used as a criteria to choose between different possible link functions, say g 1 and g 2 . Corollary 7.2. If ˆ π ( g 1 ) i < ˆ π ( g 2 ) i , we say that the estimator given under the link function g 1 is less credible than the estimator given under g 2 , that is g 2 is better than g 1 . Example 7.3. (Car Insurance Claims) The SAS Technical Report P–243 (1993) gives the following illustrative dataset of a car insurance portfolio (also reproduced in Schmitter, 2004).
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76 CHAPTER 7. CREDIBILITY FOR GLM’S Table 7.2: Car Insurance Data Class # of Risks # of Claims Car Type Age Group i = 1 500 42 small 1 2 1200 37 medium 1 3 100 1 large 1 4 400 101 small 2 5 500 73 medium 2 6 300 14 large 2 Let x i be Poisson and g be a log–link function. Here the covariates are Y i = ( y i 0 , . . . , y i 3 ) prime , where y i 0 = 1 , y i 1 = braceleftbigg 1 if car type is large 0 otherwise , y i 2 = braceleftbigg 1 if car type is medium 0 otherwise , y i 3 = braceleftbigg 1 if age group is 1 0 otherwise . In this notation Y 4 = (1 , 0 , 0 , 0) prime defines the base premium E ( X 4 ) = e β 0 for a small car type in age group 2 . The matrix of variance–covariance Σ in Proposition 7.3 is computed with SAS for weights w 1 = . . . = w 6 = 1, since here φ = 1, g ( μ ) = ln μ and V ( μ ) = μ : Σ = 0 . 008150 - 0 . 007772 - 0 . 006344 - 0 . 004623 - 0 . 007772 0 . 07418 0 . 006556 0 . 003113 - 0 . 006344 0 . 006556 0 . 01645 - 0 . 002592 - 0 . 004623 0 . 003113 - 0 . 002592 0 . 01847 . Let the tolerance level r = 0 . 1 and Y 3 = (1 , 1 , 0 , 1) prime for the third class of drivers, with a large car type in age group 1 . Then the asymptotic value in (7.7) for s 2 3 = Y prime 3 Σ Y 3 = 0 . 082236 and from (7.8) we get π 3 = Φ( ln(1+ r ) s 3 ) - Φ( ln(1 - r ) s 3 ) = 0 . 273533. Clearly, the current experience produces GLM estimators that are not credible in this class with only one claim, as s 2 3 = 0 . 082236 > 0 . 00410 = s 2 * , for r = 0 . 1 and π = 90%.
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7.3. CREDIBILITY FOR GLMS 77 By contrast, if Y 1 = (1 , 0 , 0 , 1) prime gives s 2 1 = 0 . 01737 and π 1 = 0 . 55314, which indicates a more credible GLM estimator for small cars in age group 1 than for large cars in age group 1 , although not sufficient for full credibility s 2 1 = 0 . 017374 > 0 . 00410 = s 2 * . Table 7.3 reports the asymptotic variances s 2 i = V ( Y prime ij β ) Y prime i Σ Y i and the credibility probabilities π i for all 6 classes.
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