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We consider two parametrized random digraph families, namely,proportional-edge and central similarity proximity catch digraphs (PCDs) andcompare the performance of these two PCD families in testing spatial pointpatterns. These PCD families are based on relative positions of data pointsfrom two classes and the relative density of the PCDs is used as a statisticfor testing segregation and association against complete spatial randomness.When scaled properly, the relative density of a PCD is a U-statistic. We extendthe distribution of the relative density of central similarity PCDs forexpansion parameter being larger than one. We compare the asymptoticdistribution of the statistic for the two PCD families, using the standardcentral limit theory of U-statistics. We compare finite sample performance ofthe tests by Monte Carlo simulations and prove the consistency of the testsunder the alternatives. The asymptotic performance of the tests under thealternatives is assessed by Pitman's asymptotic efficiency. We find the optimalexpansion parameters of the PCDs for testing each of the segregation andassociation alternatives in finite samples and in the limit. We demonstratethat in terms of empirical power (i.e., for finite samples) relative density ofcentral similarity PCD has better performance (which occurs for expansionparameter values larger than one) under segregation alternative, while relativedensity of proportional-edge PCD has better performance under associationalternative. The methods are illustrated in a real-life example from plantecology.

A Comparison of Two Proximity Catch Digraph Families in Testing Spatial Clustering