Statistical methods for signal estimation of point sources of cosmic rays

The estimation of the significance of the peaks in one- and two-dimensional distributions is one of the most important problems in high-energy physics and astrophysics. The physical inference from low-statistics experiments usually is biased and many discoveries lack further confirmation. One of the typical mistakes in physical inference is the use of non-adequate statistical models. We analyze the significance of the experimental evidence in the on-going efforts of detecting the point source of cosmic rays. We found that simple statistical models (Gaussian or Poisson) did not adequately describe the experimental situation of point source searches. To avoid drawbacks related to usage of the incorrect statistical model, we introduce new extremum statistical models appropriate for the point source searches. The analysis is conducted in the framework of two models utilizing extremum statistics: first - using the fixed grid of celestial coordinates, and second - using the tuned grid (introducing more degrees of freedom in the search). The test distributions for the significance estimation are obtained both from simulation models and from the analytical model of extremum statistics. We show that the second model gives adequate physical inference, while the first model can lead to the positively biased conclusions of the point source significance.