Monte Carlo tests

Do you know the test statistics of the multivariate Anderson-Darling test for multivariate normality? Don't worry, these test statistics have only been approximated for a few significance levels in terms of simulation experiments, and it is generally unknown. But then how we can estimate the value of the test statistics for a given number of observations, number of variables and a certain significance level? The answer is easy, and the procedure is as easy as for much simpler tests. We can do it with resampling methods for testing - the Monte Carlo tests.

A motivating example

Before we go to more formal descriptions of such tests, we introduce the Monte Carlo resampling test with a long introductory example. This should show why a Monte Carlo test works. The following example using body temperature data is motivated by a lecture given by Friedrich Leisch at the Vienna University of Technology and adapted for further follow-up lectures by the author of the book.

We first took...