Approximate Statistical Procedures

To carry out a statistical test, we need to know the critical value for the test statistic. In most cases this means that we must know the distribution of the test statistic under the null hypothesis. Sometimes this is known exactly, but more often only approximations are available. This may be because the distribution of the statistic is analytically intractable, or perhaps the postulated statistical model is considered only an approximation of the true underlying distributions. In both cases the use of an approximate critical value may be fully satisfactory for practical purposes.

Consider for instance the classical t-test for location. Given a sample of independent observations X1, … , Xn, we wish to test a null hypothesis concerning the mean = EX. The t-test is based on the quotient of the sample mean and the sample standard deviation Sn. If the observations arise from a normal distribution with mean then the distribution of is known exactly: It is a t-distribution with n- 1 degrees of freedom. However, we may have doubts regarding the normality, or we might even believe in a completely different model. If the number of observations is not too small, this does not matter too much. Then we may act as if possesses a standard normal distribution. The theoretical justification is the limiting result, as

provided the variables Xi have a finite second moment. This variation on the central limit theorem is proved in the next chapter. A “large sample” level test is to reject if exceeds the upper quantile of the standard normal distribution. Table 1.1 gives the significance level of this test if the observations are either normally or exponentially distributed, and