Estimation, Evaluation, and Selection of Actuarial Models

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72 CHAPTER 4. MODEL EVALUATION AND SELECTION Exponential Fit F(x) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 F n (x) p − p plot for Data Set B censored at 1,000 This plot ends at about 0.75 because that is the highest probability observed prior to the censoring point at 1,000. There are no empirical values at higher probabilities. Again, the exponential model tends to underestimate the empirical values. ¤ Exercise 77 Repeat the previous example for a Weibull model. 4.5 Hypothesis tests A picture may be worth many words, but sometimes it is best to replace the impressions conveyed by pictures with mathematical demonstrations. One such demonstration is a test of the hypotheses: H 0 : The data came from a population with the stated model. H 1 : The data did not come from such a population. The test statistic is usually a measure of how close the model distribution function is to the empirical distribution function. When the null hypothesis completely specifies the model (for example, an exponential distribution with mean 100), critical values are well-known. However, it is more often the case that the null hypothesis states the name of the model, but not its parameters. When the parameters are estimated from the data, the test statistic tends to be smaller than it would have been had the parameter values been pre-specified. That is because the estimation method itself tries to choose parameters that produce a distribution that is close to the data. In that case, the tests become approximate. Because rejection of the null hypothesis occurs for large values of the test statistic, the approximation tends to increase the probability of a Type II error while lowering the probability of a Type I error. 4 For actuarial modeling this is likely to be an acceptable trade-off. . 4 Among the tests presented here, only the chi-square test has a correction for this situation. The other tests also have corrections, but they will not be presented here.
4.5. HYPOTHESIS TESTS 73 One method of avoiding the approximation is to randomly divide the sample in half. Use one half to estimate the parameters and then use the other half to conduct the hypothesis test. Once the model is selected, the full data set could be used to re-estimate the parameters. 4.5.1 Kolmogorov-Smirnov test Let t be the left truncation point (t =0if there is no truncation) and let u be the right censoring point (u = ∞ if there is no censoring). Then, the test statistic is D = max t≤x≤u |F n(x) − F ∗ (x)| . This test should only be used on individual data. This is to ensure that the step function F n (x) is well-defined. Also, the model distribution function F ∗ (x) is assumed to be continuous over the relevant range. Example 4.14 Calculate D for Example 4.11 in the previous Section. The following table provides the needed values. Because the empirical distribution function jumps at each data point, the model distribution function must be compared both before and after the jump. The values just before the jump are denoted F n (x−) in the table. x F ∗ (x) F n (x−) F n (x) max. diff. 82 0.0391 0.0000 0.0526 0.0391 115 0.0778 0.0526 0.1053 0.0275 126 0.0904 0.1053 0.1579 0.0675 155 0.1227 0.1579 0.2105 0.0878 161 0.1292 0.2105 0.2632 0.1340 243 0.2138 0.2632 0.3158 0.1020 294 0.2622 0.3158 0.3684 0.1062 340 0.3033 0.3684 0.4211 0.1178 384 0.3405 0.4211 0.4737 0.1332 457 0.3979 0.4737 0.5263 0.1284 680 0.5440 0.5263 0.5789 0.0349 855 0.6333 0.5789 0.6316 0.0544 877 0.6433 0.6316 0.6842 0.0409 974 0.6839 0.6842 0.7368 0.0529 1,193 0.7594 0.7368 0.7895 0.0301 1,340 0.7997 0.7895 0.8421 0.0424 1,884 0.8983 0.8421 0.8947 0.0562 2,558 0.9561 0.8947 0.9474 0.0614 3,476 0.9860 0.9474 1.0000 0.0386 The maximum occurs at D =0.1340. For Data Set B censored at 1,000, 15 of the 20 observations are uncensored. The following table illustrates the needed calculations.
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