# 16.4 Hypothesis tests

A picture may be worth many words, but sometimes it is best to replace the impressions conveyed by pictures with mathematical demonstrations.^{4} One such demonstration is a test of the following 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 (e.g., 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 prespecified. This relationship occurs because the estimation method itself tries to choose parameters that produce a distribution that is close to the data. When parameters are estimated from data, 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 (declaring the model is acceptable when it is not) while lowering the probability of a Type I error (rejecting an acceptable model).^{5} For actuarial modeling this tendency is likely to be an acceptable trade-off.

One method of avoiding the approximation ...

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