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We consider a special type of interval catch digraph (ICD) family forone-dimensional data in a randomized setting and propose its use for testinguniformity. These ICDs are defined with an expansion and a centralityparameter, hence we will refer to this ICD as parameterized ICD (PICD). Wederive the exact (and asymptotic) distribution of the domination number of thisPICD family when its vertices are from a uniform (and non-uniform) distributionin one dimension for the entire range of the parameters; thereby determine theparameters for which the asymptotic distribution is non-degenerate. We observejumps (from degeneracy to non-degeneracy or from a non-degenerate distributionto another) in the asymptotic distribution of the domination number at certainparameter combinations. We use the domination number for testing uniformity ofdata in real line, prove its consistency against certain alternatives, andcompare it with two commonly used tests and three recently proposed tests inliterature and also arc density of this ICD and of another ICD family in termsof size and power. Based on our extensive Monte Carlo simulations, wedemonstrate that domination number of our PICD has higher power for certaintypes of deviations from uniformity compared to other tests.

Domination Number of an Interval Catch Digraph Family and its use for Testing Uniformity