Estimation, Evaluation, and Selection of Actuarial Models

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80 CHAPTER 4. MODEL EVALUATION AND SELECTION 1. Use a simple model if at all possible. 2. Restrict the universe of potential models. The methods outlined in the remainder of this Section will help with the first point. The second one requires some experience. Certain models make more sense in certain situations, but only experience can enhance the modeler’s senses so that only a short list of quality candidates are considered. The Section is split into two types of selection criteria. The first set is based on the modeler’s judgment while the second set is more formal in the sense that most of the time all analysts will reach the same conclusions. That is because the decisions are made based on numerical measurements rather than charts or graphs. Exercise 86 (*) One thousand policies were sampled and the number of accidents for each recorded. The results are in the table below. Without doing any formal tests, determine which of the following five models is most appropriate: binomial, Poisson, negative binomial, normal, gamma. No. of accidents No. of policies 0 100 1 267 2 311 3 208 4 87 5 23 6 4 Total 1000 4.6.2 Judgment-based approaches Using one’s own judgment to select models involves one or more of the three concepts outlined below. In both cases, the analyst’s experience is critical. First, the decision can be based on the various graphs (or tables based on the graphs) presented in this Chapter. 5 This allows the analyst to focus on aspects of the model that are important for the proposed application. For example, it may be more important to fit thetailwell,oritmaybe more important to match the mode or modes. Even if a score-based approach is used, it may be appropriate to present a convincing picture to support the chosen model. Second, the decision can be influenced by the success of particular models in similar situations or the value of a particular model for its intended use. For example, the 1941 CSO mortality table follows a Makeham distribution for much of its range of ages. In a time of limited computing power, such a distribution allowed for easier calculation of joint life values. As long as the fit of this model was reasonable, this advantage outweighed the use of a different, but better fitting, model. Similarly, if the Pareto distribution has been used to model a particular line of liability insurance both by the analyst’s company and by others, it may require more than the usual amount of evidence to change to an alternative distribution. 5 Besides the ones discussed in this Note, there are other plots/tables that could be used. Other choices are a q − q plot, and a comparison of model and empirical limited expected value or mean residual life functions.
4.6. SELECTING A MODEL 81 Third, the situation may completely determine the distribution. For example, suppose a dental insurance contract provides for at most two check-ups per year and suppose that individuals make two independent choices each year as to whether or not to have a check-up. If each time the probability is p, then the distribution must be binomial with n =2. Finally, it should be noted that the more algorithmic approaches outlined below do not always agree. In that case judgment is most definitely required, if only to decide which algorithmic approach to use. 4.6.3 Score-based approaches Some analysts might prefer an automated process for selecting a model. An easy way to do that would be to assign a score to each model and the model with the highest or lowest value wins. The following scores are worth considering: 1. Lowest value of the Kolmogorov-Smirnov test statistic 2. Lowest value of the Anderson-Darling test statistic 3. Lowest value of the chi-square goodness-of-fit test statistic 4. Highest p-value for the chi-square goodness-of-fit test 5. Highest value of the likelihood function at its maximum. All but the p-value have a deficiency with respect to parsimony. First, consider the likelihood function. When comparing, say, an exponential to a Weibull model, the Weibull model must have a likelihood value that is at least as large as the exponential model. They would only be equal in the rare case that the maximum likelihood estimate of the Weibull parameter τ is equal to 1. Thus, the Weibull model would always win over the exponential model, a clear violation of the principle of parsimony. For the three test statistics, there is no assurance that the same relationship will hold, but it seems likely that if a more complex model is selected, the fit measure is likely to be better. Theonlyreasonthep-value is immune from this problem is that with more complex models the distribution used has fewer degrees of freedom. It is then possible that the more complex model will have a smaller p-value. There is no comparable adjustment for the first two test statistics listed. With regard to the likelihood value, there are two ways to proceed. One is to perform the likelihood ratio test and the other is to extract a penalty for employing additional parameters. The likelihood ratio test is technically only available when one model is a special case of another (for example, Pareto vs. generalized Pareto). The concept can be turned into an algorithm by using the test at a 5% significance level. Begin with the best one-parameter model (the one with the highest loglikelihood value). Add a second parameter only if the two-parameter model with the highest loglikelihood value shows an increase of at least 1.92 (so twice the difference exceeds the critical value of 3.84). Then move to three-parameter models. If the comparison is to a two-parameter model, a 1.92 increase is again needed. If the early comparison led to keeping the one-parameter model, an increase of 3.00 is needed (because the test has two degrees of freedom). To add three parameters requires a 3.91 increase, four parameters a 4.74 increase, and so on. In the spirit of this Chapter, this algorithm can be used even for non-special cases. However, it would not be appropriate to claim that a likelihood ratio test was being conducted.
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