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Statistical pattern classification methods based on data-random graphs wereintroduced recently. In this approach, a random directed graph is constructedfrom the data using the relative positions of the data points from variousclasses. Different random graphs result from different definitions of theproximity region associated with each data point and different graph statisticscan be employed for data reduction. The approach used in this article is basedon a parameterized family of proximity maps determining an associated family ofdata-random digraphs. The relative arc density of the digraph is used as thesummary statistic, providing an alternative to the domination number employedpreviously. An important advantage of the relative arc density is that,properly re-scaled, it is a $U$-statistic, facilitating analytic study of itsasymptotic distribution using standard $U$-statistic central limit theory. Theapproach is illustrated with an application to the testing of spatial patternsof segregation and association. Knowledge of the asymptotic distribution allowsevaluation of the Pitman and Hodges-Lehmann asymptotic efficacies, andselection of the proximity map parameter to optimize efficiency. Furthermorethe approach presented here also has the advantage of validity for data in anydimension.

Relative Density of the Random $r$-Factor Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association