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We consider the distribution of a graph invariant of central similarityproximity catch digraphs (PCDs) based on one dimensional data. The centralsimilarity PCDs are also a special type of parameterized random digraph familydefined with two parameters, a centrality parameter and an expansion parameter,and for one dimensional data, central similarity PCDs can also be viewed as atype of interval catch digraphs. The graph invariant we consider is therelative density of central similarity PCDs. We prove that relative density ofcentral similarity PCDs is a U-statistic and obtain the asymptotic normalityunder mild regularity conditions using the central limit theory ofU-statistics. For one dimensional uniform data, we provide the asymptoticdistribution of the relative density of the central similarity PCDs for theentire ranges of centrality and expansion parameters. Consequently, wedetermine the optimal parameter values at which the rate of convergence (tonormality) is fastest. We also provide the connection with class cover catchdigraphs and the extension of central similarity PCDs to higher dimensions.

Distribution of the Relative Density of Central Similarity Proximity Catch Digraphs Based on One Dimensional Uniform Data