A likelihood ratio test based on the Poisson point model in clustering problems

When testing the presence of (k + 1) clusters versus the presence of k clusters,
Hardy (1983) considers a stationary Poisson point process in some domain D C IR? which is the union of k disjoint convex compact domains D; (i = 1,2,---,k) (k fixed). In order to derive a stopping rule for determining the ’optimal’ number of clusters present in a given set of data, Hardy (1983) [5] proposed the likelihood ratio test for Hy : v = & versus H, : v = k+1. However, one can see that & (the number of components) is not a parameter of the model. The goal of this small note is to give a more accurate formulation of this test, which is based on the concept of finite mixture models (see Redner and Walker (1984) [9], Izenman and Sommer (1988)[3]).