  Statistical pattern classification methods based on data-random graphs were
introduced recently. In this approach, a random directed graph is constructed
from the data using the relative positions of the data points from various
classes. Different random graphs result from different definitions of the
proximity region associated with each data point and different graph statistics
can be employed for data reduction. The approach used in this article is based
on a parameterized family of proximity maps determining an associated family of
data-random digraphs. The relative arc density of the digraph is used as the
summary statistic, providing an alternative to the domination number employed
previously. An important advantage of the relative arc density is that,
properly re-scaled, it is a $U$-statistic, facilitating analytic study of its
asymptotic distribution using standard $U$-statistic central limit theory. The
approach is illustrated with an application to the testing of spatial patterns
of segregation and association. Knowledge of the asymptotic distribution allows
evaluation of the Pitman and Hodges-Lehmann asymptotic efficacies, and
selection of the proximity map parameter to optimize efficiency. Furthermore
the approach presented here also has the advantage of validity for data in any
dimension.
