On some significance tests in cluster analysis

We investigate the properties of several significance tests for distinguishing between the hypothesisH of a homogeneous population and an alternativeA involving clustering or heterogeneity, with emphasis on the case of multidimensional observationsx₁, ...,x_n ^p. Four types of test statistics are considered: the (s-th) largest gap between observations, their mean distance (or similarity), the minimum within-cluster sum of squares resulting from a k-means algorithm, and the resulting maximum F statistic. The asymptotic distributions underH are given forn and the asymptotic power of the tests is derived for neighboring alternatives.