Domination number of an interval catch digraph family and its use for testing uniformity

Abstract

We consider a special type of interval catch digraph (ICD) for one-dimensional data in a randomized setting and propose its use for testing uniformity. These ICDs are defined with expansion and centrality parameters, hence are called parameterized ICDs (PICDs). We derive the exact (and asymptotic) distribution of the domination number of this PICD when its vertices are from a uniform (and non-uniform) distribution in one dimension for the entire parameter ranges; thereby determine the parameters for which the asymptotic distribution is non-degenerate. We use the domination number for testing uniformity of one-dimensional data, prove its consistency against certain alternatives, and compare it with commonly used and recently proposed tests in literature and also arc density of two ICD families in terms of size and power. Based on our Monte Carlo simulations, we demonstrate that PICD domination number has higher power for certain types of alternatives compared to other tests.