Estimation, Evaluation, and Selection of Actuarial Models

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74 CHAPTER 4. MODEL EVALUATION AND SELECTION x F ∗ (x) F n (x−) F n (x) max. diff. 27 0.0369 0.00 0.05 0.0369 82 0.1079 0.05 0.10 0.0579 115 0.1480 0.10 0.15 0.0480 126 0.1610 0.15 0.20 0.0390 155 0.1942 0.20 0.25 0.0558 161 0.2009 0.25 0.30 0.0991 243 0.2871 0.30 0.35 0.0629 294 0.3360 0.35 0.40 0.0640 340 0.3772 0.40 0.45 0.0728 384 0.4142 0.45 0.50 0.0858 457 0.4709 0.50 0.55 0.0791 680 0.6121 0.55 0.60 0.0621 855 0.6960 0.60 0.65 0.0960 877 0.7052 0.65 0.70 0.0552 974 0.7425 0.70 0.75 0.0425 1000 0.7516 0.75 0.75 0.0016 The maximum occurs at D =0.0991. ¤ All that remains is to determine the critical value. Commonly used critical values for this test are 1.22/ √ n for α =0.10, 1.36/ √ n for α =0.05, and1.63/ √ n for α =0.01. When u0. Exercise 80 (*) You are given the following five observations from a random sample: 0.1, 0.2, 0.5, 1.0, and 1.3. Calculate the Kolmogorov-Smirnov test statistic for the null hypothesis that the population density function is f(x) =2(1+x) −3 , x>0.
4.5. HYPOTHESIS TESTS 75 4.5.2 Anderson-Darling test This test is similar to the Kolmogorov-Smirnov test, but uses a different measure of the difference between the two distribution functions. The test statistic is Z u A 2 [F n (x) − F ∗ (x)] 2 = n t F ∗ (x)[1 − F ∗ (x)] f ∗ (x)dx. That is, it is a weighted average of the squared differences between the empirical and model distribution functions. Note that when x is close to t or to u, the weights might be very large due to the small value of one of the factors in the denominator. This test statistic tends to place more emphasis on good fit in the tails than in the middle of the distribution. Calculating with this formula appears to be challenging. However, for individual data (so this is another test that does not work for grouped data), the integral simplifies to kX A 2 = −nF ∗ (u)+n [1 − F n (y j )] 2 {ln[1 − F ∗ (y j )] − ln[1 − F ∗ (y j+1 )]} j=0 kX +n F n (y j ) 2 [ln F ∗ (y j+1 ) − ln F ∗ (y j )] j=1 where the unique non-censored data points are t = y 0
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