Random dendrograms for classifiability testing

We propose statistical tests to decide between an hypothesis of non classifiability of the data against the presence of a classification in some simple situations. We consider only the single link algorithm and two null hypotheses of non classifiability, according to whether the dissimilarities or the objects themselves are i.i.d. random variables. Each choice for the distribution of the input of the single link algorithm induces a different probability distribution on the output, which is a random indexed dendrogram. Certain characteristics of these random indexed dendrograms are studied, and their asymptotic distributions computed under each null hypothesis. All these random variables can be used to define statistical tests. Explicit examples of such tests are provided.