On testing for clusters using the sample covariance

This paper analyzes the problem of using the sample covariance matrix to detect the presence of clustering in p-variate data in the special case when the component covariance matrices are known up to a constant multiplier. For the case of testing one population against a mixture of two populations, tests are derived and shown to be optimal in a certain sense. Some of their distribution properties are derived exactly. Some remarks on the extensions of these tests to mixtures of k <= p populations are included. The paper is essentially a formal treatment (in a special case) of some well-known procedures. The methods used in deriving the distribution properties are applicable to a variety of other situations involving mixtures.