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.

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