Estimation, Evaluation, and Selection of Actuarial Models

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iv CONTENTS 4.5.2 Anderson-Darlingtest ............................... 75 4.5.3 Chi-square goodness-of-fit test .......................... 77 4.6 Selectingamodel...................................... 79 4.6.1 Introduction .................................... 79 4.6.2 Judgment-basedapproaches............................ 80 4.6.3 Score-basedapproaches .............................. 81 5 Models with covariates 85 5.1 Introduction......................................... 85 5.2 Proportionalhazardsmodels ............................... 86 5.3 Thegeneralizedlinearmodel ............................... 92 A Solutions to Exercises 95 B Using Microsoft Excel TM 131
Chapter 1 Introduction For some time, the subjects of “Loss Models” and “Survival Models” have been part of the actuarial syllabus. The former was most often associated with property and liability insurances while the latter was most often associated with life and disability insurances. However, the goal of these two subjects is the same — the determination of models for random phenomena. The purpose of this Study Note is to present the two subjects in a unified context that indicates that either approach may be appropriate for a given actuarial problem. The difference between Parts 3 and 4 1 is reflected in the course names. Part 3 is “actuarial models” and the objective is to be able to work with the most common actuarial models. Most such models are random variables, such as the time to death or the amount of an automobile physical damage claim. After mastering the content of Part 3, the student is left wondering where the models come from. That is the purpose of Part 4, “actuarial modeling.” One option is to simply announce the model. Your boss needs a model for basic dental payments. You announce that it is the lognormal distribution with µ =5.1239 and σ =1.0345 (the many decimal places are designed to give your announcement an aura of precision). When your boss, or a regulator, or an attorney who has put you on the witness stand, asks you how you know that to beso,itwilllikelynotbesufficient to answer that “I just know these things.” It may not even be sufficient to announce that your actuarial friend at Gamma Dental uses that model. An alternative is to collect some data and use it to formulate a model. Most distributional models have two components. The first is a name, such as “Pareto.” The second is the set of values of parameters that complete the specification. Matters would be simpler if modeling could be done in that order. Most of the time we need to fix the parameters that go with a named model before we can decide if we want to use that type of model. Thus, this note begins with parameter estimation, the subject of Chapter 2. By the end of that Chapter, we will have a variety of estimation methods at our disposal. In Chapter 3 we learn how to evaluate the various methods and in particular learn how to measure the accuracy of our estimator. Chapter 4 is devoted to model selection. Two basic questions are answered. The first one is “Is our model acceptable for use?” while the second one is “From our set of acceptable models, which one should we use?” Chapter 5 introduces some new models that were not covered in Part 3. 1 The Casualty Actuarial Society refers to its education units as “exams” while the Society of Actuaries refers to them as “courses.” To avoid making a decision, this Note will refer to units as “parts,” a term in favor when the author was going through this process. 1
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Estimation, Evaluation, and Selection of Actuarial Models

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