Two-Stage Tests Of Hypotheses For Principal Components

A simulation study of an asymptotic test for a specific principal component (Anderson, 1963) indicates that for small sample size the actual probability of type I error may be quite a bit larger than the nominal significance level. In this paper we consider a two-stage test of the more general hypothesis for the space spanned by the set of the first p principal components. The first test in this two-stage procedure is an exact test due to Mallows (1961), while the second test is a conservative test obtained via the union-intersection principle. This two-stage test offers an alternative method to the asymptotic test, whereby the probability of a type I error can be reasonably controlled regardless of the sample size