Tests for simultaneously determining numbers of clusters and their shape with multivariate data

Given a set of data, very little is known about tests to determine number of clusters and/or elements of the clusters. Even in the simplest case of detecting between only one or two clusters with multivariate normal data, theoretically the number of tests needed seems to be infinite. Alternatively, suppose N independent estimates of generalized variances (GVs) are computed from a given set of p -dimensional vector observations. Assuming multivariate normality, tests based on GVs are proposed which objectively and uniquely determine, simultaneously, the number of clusters and their corresponding elements. Only a reasonably small nunber of tests are required for this stepwise procedure. The exact percentage points are either available from existing tables or can be computed from a result presented.