Estimation, Evaluation, and Selection of Actuarial Models

72 CHAPTER 4. MODEL EVALUATION AND SELECTION
Exponential Fit
F(x)
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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.
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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.
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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.