We develop the statistical methods for comparing two sets of arrival
directions of cosmic rays in which the two-dimensional distribution of arrival
directions is reduced to the one-dimensional distributions so that the standard
one-dimensional Kolmogorov-Smirnov test can be applied. Then we apply them to
the analysis of correlation between the ultra-high energy cosmic rays (UHECR)
with energies above 5.7×1019 eV, observed by Pierre Auger Observatory
(PAO) and Akeno Giant Air Shower Array (AGASA), and the active galactic nuclei
(AGN) within the distance 100 Mpc. For statistical test, we set up the simple
AGN model for UHECR sources in which a certain fraction of observed UHECR are
originated from AGN within a chosen distance, assuming that all AGN have equal
luminosity and smearing angle of UHECR, and the remaining fraction are from the
isotropic background contribution. For the PAO data, our methods exclude not
only a hypotheses that the observed UHECR are simply isotropically distributed
but also a hypothesis that they are completely originated from the selected
AGN. But, the addition of appropriate amount of isotropic component either
through the background contribution or through the large smearing effect
improves the correlation greatly and makes the AGN hypothesis for UHECR sources
a viable one. We also point out that restricting AGN within the distance bin of
40-60 Mpc happens to yield a good correlation without appreciable isotropic
component and large smearing effect. For the AGASA data, we don't find any
significant correlation with AGN.Comment: 18 pages, 9 figures, a version published in JCA
