  The astrophysical origins of the majority of the IceCube neutrinos remain
unknown. Effectively characterizing the spatial distribution of the neutrino
samples and associating the events with astrophysical source catalogs can be
challenging given the large atmospheric neutrino background and underlying
non-Gaussian spatial features in the neutrino and source samples. In this
paper, we investigate a framework for identifying and statistically evaluating
the cross-correlations between IceCube data and an astrophysical source catalog
based on the $k$-Nearest Neighbor Cumulative Distribution Functions
($k$NN-CDFs). We propose a maximum likelihood estimation procedure for
inferring the true proportions of astrophysical neutrinos in the point-source
data. We conduct a statistical power analysis of an associated likelihood ratio
test with estimations of its sensitivity and discovery potential with synthetic
neutrino data samples and a WISE-2MASS galaxy sample. We apply the method to
IceCube's public ten-year point-source data and find no statistically
significant evidence for spatial cross-correlations with the selected galaxy
sample. We discuss possible extensions to the current method and explore the
method's potential to identify the cross-correlation signals in data sets with
different sample sizes.
