Estimation, Evaluation, and Selection of Actuarial Models

62 CHAPTER 4. MODEL EVALUATION AND SELECTION
This Chapter begins with a review of hypothesis testing, mostly because it is no longer directly
covered on prior actuarial examinations. The remaining sections cover a variety of evaluation and
selection tools. Each tool has its own strengths and weaknesses and it is possible for different tools
to lead to different models. This makes modeling as much art as science. At times, in real-world
applications, the model’s purpose may lead the analyst to favor one tool over another.
4.2 A review of hypothesis tests
The Neyman-Pearson approach to hypothesis testing is covered in detail in most mathematical
statistics texts. This review will be fairly straightforward and will not address philosophical issues
or consider alternative approaches. A hypothesis test begins with two hypotheses, one called the
null and one called the alternative. The traditional notation is H 0 for the null hypothesis and H 1
for the alternative hypothesis. The two hypotheses are not treated symmetrically. Reversing them
may alter the results. To illustrate this process, a simple example will be used.
Example 4.1 Your company has been basing its premiums on an assumption that the average
claim is 1,200. You want to raise the premium and a regulator has demanded that you provide
evidence that the average now exceeds 1,200. To provide such evidence, the numbers in Data Set B
have been obtained. What are the hypotheses for this problem?
Let µ be the population mean. One hypothesis (the one you claim is true) is that µ>1, 200.
Because hypothesis tests must present an either/or situation, the other hypothesis must be µ ≤
1, 200. The only remaining task is to decide which of them is the null hypothesis. Whenever the
universe of continuous possibilities is divided in two there is likely to be a boundary that needs to
be assigned to one hypothesis or the other. The hypothesis that includes the boundary must be
the null hypothesis. Therefore, the problem can be succinctly stated as:
H 0 : µ ≤ 1, 200
H 1 : µ>1, 200.
¤
The decision is made by calculating a quantity called a test statistic. It is a function of the
observations and is treated as a random variable. That is, in designing the test procedure we are
concerned with the samples that might have been obtained and not with the particular sample
that was obtained. The test specification is completed by constructing a rejection region. Itisa
subset of the possible values of the test statistic. If the value of the test statistic for the observed
sample is in the rejection region, the null hypothesis is rejected and the alternative hypothesis
is announced as the result that is supported by the data. Otherwise, the null hypothesis is not
rejected (more on this later). The boundaries of the rejection region (other than plus or minus
infinity) are called the critical values.
Example 4.2 For the continuing example, complete the test using the test statistic and rejection
region that is promoted in most statistics books. Assume that the population that produced Data Set
B has a normal distribution with standard deviation 3,435.